MAT0024C

Beginning Algebra

Class Notes

Mr. Groccia

MAT0024C Beginning Algebra

Chapter 1: Real Numbers

Chapter 2: Algebraic Expressions

and Equations

Chapter 3: Graphing

Chapter 4: Polynomials

Chapter 5: Factoring

Chapter 6: Radicals

Chapter 1: Real Numbers

1.1: Displaying Information and Vocabulary

1.2: Fractions

1.3: Real Numbers

1.4: Adding/ Subtracting Real Numbers

1.5: Multiplying/ Dividing Real Numbers

1.6: Exponents and Order of Operations

1.7: Algebraic Expressions

1.1: Displaying Information and Vocabulary

Why do we need Tables and Graphs?

Creating a table

Table Title: Movies Box Office Sales

Source: www.leesmovieinfo.com

Column Title for Bar Graph: Movie Title	Column Title: Box Office Sales (US and Canada) Rounded
Shrek	$268,000,000
The Notebook	$80,000,000
Star Wars	$461,000,000
Happy Gilmore	$39,000,000
The Godfather	$135,000,000
Rocky	$117,000,000

Creating a Bar Graph

Vertical Axis

Horizontal Axis

Creating a Line Graph

Table for Line Graph Sales for Luxury Car Dealer in United States during 2001
Month	Sales
January	$2,500,000
June	$6,250,000
September	$750,000
December	$9,000,000

Vertical Axis

Horizontal Axis

Why do we need Tables and Graphs?

__________________________________________________________

Vocabulary

Sum- is the result of an addition.

Example: The sum of 5 and 7 is 12

Difference- is the result of a subtraction.

Example: The difference of 3 and 2 is 1

Product- is the result of a multiplication.

Example: The product of 4 and 7 is 28.

Quotient- is the result of a division.

Example: The quotient of 12 and 3 is 4.

Notation

Multiplication Symbols

Times Sign (Will not be used often due to “” being used as a variable)

Example: 54 = 20

Raised dot

Example: 32 = 6

Parentheses

Example:

Division Symbols

Division Sign

Example: 153 = 5

Long Division

Example:

Fraction Bar

Example: .

Variables, Expression, and Equations

Variables- Letters that stand for numbers.

Example: If you don’t know how much money you have it can represented with an “x”.

Equation- a mathematical sentence that contains an = symbol.

Examples: 3 + 5 = 8 or x + 9 = 12

Algebraic Expression- variables and/or numbers that can be combined with the operations of addition, subtraction, multiplication, and division.

Examples: x + 7 or or

Constructing Tables

Given the situation:

Movies cost $8 each.

How much would it cost to bring my friends to the movies?

We don’t know how many friends.

We will use the variable “f” to represent friends.

We also don’t know the total cost.

We will use the variable “c” to represent total cost.

What will the formula be to represent this situation?

_________________________________________

Use this table to show possible costs based on how many friends go to the movies:

Cost of going to the movies

f	c
1	8

Section 1.1: Intro Language of Algebra Practice Problems

1. Create a table and bar graph for the following information:

Here are most played songs of 2006: (Source: www.nielsenmedia.com)

“Be Without You” Mary J. Blige (395,995 times), “Unwritten” Natasha Bedingfield (336,276)
“Temperature” Sean Paul (324,555), “Me & U” Cassie (312,073)
“Hips Don't Lie” Shakira (308,903), “Promiscuous” Nelly Furtado (292,264)

TABLE: BAR GRAPH:

2. Create a table and line graph for the following information:

Here are the yearly enrollemnt for Valencia College:

(Source: http://valenciacollege.edu/IR/EnrollmentStatistics.cfm)

1967: 567 1977: 24,483 1987: 54,515 1997: 48,503 2005: 53,806

TABLE: LINE GRAPH:

3. Match the vocabulary word to the definition or symbols:

_____ 1. Algebraic Expression a. Result of addition

_____ 2. Product b. Letters that stand for numbers

_____ 3. Division Symbols c. Result of division

_____ 4. Quotient d. Algebraic equation for the cost of 3 unknown priced tickets with a coupon for $2 off is $10.

_____ 5. 3(x) – 2 = 10 e. Result of subtraction

_____ 6. Difference f. Variables and/or numbers combined with arithmetic operations

_____ 7. Multiplication Symbols g. , ,

_____ 8. Variables h. , ,

_____ 9. Sum i. Result of multiplication

_____ 10. Equation j. Mathematical Sentence with an =

1.2: Fractions

Factor- means to express as a product of two or more numbers.

Ex. or or or

Prime number- is a whole number greater than 1 that has only itself and 1 as factors.

The first ten prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29

Composite number- is a whole number greater than 1 that is not prime.

The first ten composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16 and 18

Prime Factorization- every composite number can be factored into the product of two or more prime numbers.

Example: Find the prime factorization of 210

		210
	21		10

3	7	2	5

The prime factors are:

Practice Examples:

Find the prime factorization of 256:

Find the prime factorization of 189:

What is a FRACTION and why do we need them?

Meaning of Fractions

Example of a fraction:

Numerator

Fraction bar

Denominator

Special Fraction Forms: For any nonzero number a,

ex.

Simplifying Fractions- a fraction is in simplest form, or lowest terms, when the numerator and denominator have no common factors other

Example: Simplify

Solution:

Find the prime factorization of the numerator and denominator. If the numerator and denominator have common factors, those factors become one.

Practice Examples:

Simplify:

Mixed Numbers- represents the sum of a whole number and a fraction.

Ex.

Improper Fraction- a fraction where the numerator greater than or equal to denominator. Ex.

Converting from an improper fraction to a mixed fraction:

Example:

Solution:

Multiply the denominator by the whole number, and then add the numerator. That number becomes the numerator with the original denominator.

Practice Examples:

Convert from mixed fraction to an improper fraction:

Converting from an improper fraction to a mixed fraction:

Example:

Solution:

Divide the denominator into the numerator. The result will be the whole number, the remainder will be the numerator, and the original denominator will be the denominator.

Practice Examples:

Convert from improper fraction to a mixed fraction:

Multiplying Fractions- to multiply fractions, multiply the numerators and multiply the

denominators.

Example: Multiply Remember to simplify your answer if needed.

Solution:

Practice Examples:

Multiply:

Dividing Fractions- to divide fractions, multiply the first fraction by the reciprocal of the second. Remember to simplify your answer.

Example: Multiply Remember to simplify your answer if needed.

Solution:

Practice Examples:

Divide:

Adding and Subtracting Fractions- in order to add or subtract fractions, they must have the same denominator.

To add (or subtract) two fractions with same denominator, add (or subtract) their numerators and write the sum (or difference) over the common denominator.

Example:

Solutions:

To add (or subtract) two fractions with unlike denominator, find the least common denominator, convert both fractions with the common denominator, then add (or subtract) their numerators and write the sum (or difference) over the common denominator.

Least Common Denominator (LCD)- for a set of fractions is the smallest number each denominator will denominator will divide exactly (divide with no remainder).

To find the LCD, find the prime factorization of both denominators and use each prime factor the greatest number of times it appears in any one factorization.

Example: Find the LCD of 10 and 28

Solution: First find the prime factorization of each number:

Then use each factor the greatest amount of times in each factor:

Practice Examples:

Find the LCD of 15 and 20 Find the LCD of 8 and 12

Adding and Subtracting Fractions with unlike denominators

Example:

Solutions:

The LCD is: The LCD is:

Practice Examples:

Solve:

What is a FRACTION and why do we need them?

____________________________________________________________________________________________________________________________________________________________

Section 1.2: Fractions Practice Problems

1. Find the prime factorization of 184: 2. Find the prime factorization of 212:

________________________________ __________________________________

3. Simplify: _____ 4. Simplify: ______ 5. Simplify: _____

6. Simplify: _____ 7. Simplify: _____ 8. Simplify: _____

Convert from mixed fraction to an improper fraction:

9. _____ 10. _____ 11. ______

Convert from improper fraction to a mixed fraction:

12. _____ 13. _____ 14. _____

Section 1.2: Fractions Practice Problems Continue

Multiply:

15. _____ 16. _____ 17. _____

Divide:

18. _____ 19. _____ 20. _____

21. Find LCD of 18 and 24 22. Add: _____ 23. Subtract: _____

Solve:

24. _____ 25. _____ 26. ______

1.3: Real Numbers

What are REAL NUMBERS?

Set- collection of numbers, the symbol used is:

Natural Numbers-

Whole Numbers-

Integers-

Rational Numbers- any number that can be written as a fraction with integer numerator and nonzero integer denominator.

Examples:

Irrational Number- nonterminating, nonrepeating decimal

Examples:

Real Number- rational and irrational numbers, all points on the number line.

Real Number V V
Rational Number V	Irrational Number
Integer V
Whole Number V
Natural Number

Classify the following numbers:

7
-5

Real Number Line

pic_95472

Graph the following on the number line:

Math Symbols

< Less Than > Greater Than

Fill in the blanks with < or >

-4		4
-2		-3
-5		-4
1.09		1.1

Opposite- 2 numbers that are the same distance from O. Symbol: ( - )

Example: 4 and -4 -3 and 3

Absolute Value: the distance from 0. Symbol:

Example:

Fill in the blanks with < or > or =

		4
-(-5)		-3
-		10
-(9)		8

What are REAL NUMBERS?

__________________________________________________________

Section 1.3: Real Numbers Practice Problems

1. Classify the following numbers as: Natural, Whole, Integers, Rational, Irrational, Real

Numbers may fall in more than 1 category.

a. _________________________________________________________________

b. -10 _________________________________________________________________

c. _________________________________________________________________

d. 23.458976975…. ___________________________________________________________

e. 0 _________________________________________________________________

f. _________________________________________________________________

2. Create a number line and graph the following on the number line:

3. Complete the table with: < Less Than OR > Greater Than OR = Equal to

a.	-17	-18
b.	3.001	3.01
c.
d.
e.	-(-2)	-(2)
f.	-	-
g.		-
h.
i.	-(0.003)	-

1.4: Adding/Subtracting Real Numbers

How do you ADD REAL NUMBERS?

Adding Real Numbers

Signed Numbers- positive and negative numbers

pic_95472

Adding Two Numbers with the Same Sign

1. Two positive numbers add them and keep the sign.

Example: 10 + 3 = 13 15 + 12 = 27

2. Two negative numbers add them and keep the sign.

Example: -2 + (-3) = -5 -1 + (-7) = -8

Practice Examples:

a) -22 + (-13) =

b) -1.23 + (-2.45) =

Adding Two Numbers with Different Signs

Subtract the numbers and take the sign of the larger number.

(Larger number without looking at the sign of the number)

Example: -5 + 2 = -3 4 + (-1) = 3

Practice Examples:

a) -30 + 10 =

b) 5.4 + (-2.2) =

d) -20 + 5 + (-12) + (-3) + 7 =

e) (-7 + 8) + 2 + (-13 + 12) =

Properties of Addition

Commutative Property of Addition- changing the order when adding does not affect

the answer

Example: 2 + 3 = 3 + 2

Associative Property of Addition- changing the grouping when adding does not affect

the answer

Example: (2 + 3) + 4 = 2 + (3 + 4)

Addition Property of 0- when 0 is added to any real number, the result is the same as

real number.

Example: 5 + 0 = 5

Addition Property of Opposites- the sum of a number and its opposite is 0.

Example: 7 + (-7) = 0

How do you ADD REAL NUMBERS?

______________________________________________________________

How do you SUBTRACT REAL NUMBERS?

Subtracting Real Numbers

Opposite of an opposite- is the original number

Example: - (-5) is 5

Practice Examples:

a. - (-7) = b. - (-g) = c. =

Subtraction of Real Numbers

Change the subtraction sign to an addition sign and take the opposite of the following number. Then follow addition rules.

Example: -5 – 3 =

Solution:

Change the subtraction sign to addition and take the opposite of the following number.

-5 – 3 =

-5 + (-3) =

Then follow addition rules.

-5 + (-3) = -8

_______________________________________________________________________

Example: 7 – (-9) =

Solution:

Change the subtraction sign to addition and take the opposite of the following number.

7 – (-9) =

7 + 9 =

Follow addition rules.

7 + 9 = 16

Practice Examples:

a) -10 - 7 = b) 50 – 85 =

c) = d) -7 – 4 + 10 – (-5) =

e) Water level. In one week, the water level in a storage tank went from 25 feet above normal to 12 feet below. Find the change in the water level.

How do you SUBTRACT REAL NUMBERS?

______________________________________________________________

Section 1.4:Adding/Subtracting Real Numbers Practice Problems

Solve the following problems

1a. 10 + (-2) = ________ b. -4 + (-3) = ________ c. -12 + 6 = ________

2a. -3.4 + (-2.15) = ________ b. ________ c. ______

3a. -5 – 5 = ________ b. 10 - (-4) = ________ c. -3 – (-8) = ________

4a. 6.25 – 10.75 = ________ b. ________ c. ____

5a. 7 + (-2) + (-3) + 8 + (-3) = ________ b. -2.3 + (-4.2) + 1.24 + (-0.4) = _______

6a. 15 - (-3) + (-6) - 21 - (-1) = ________ b. __________

7. Sam is on the fifth floor of a building and needs to use the restroom that is located on the third floor of the basement (3 floors below ground level). How many floors must he travel to use the restroom? Give the equation and the solution to this problem.

1.5: Multiplying and Dividing Real Numbers

How do you MULTIPLY/DIVIDE REAL NUMBERS?

Multiplying or Dividing Real Numbers

If the signs are the same, the solution is positive.

Examples:

If the signs are different, the solution is negative.

Examples:

Practice Examples:

Properties of Multiplication

Commutative Property of Multiplication- changing the order when multiplying does

not affect the answer.

Example:

Associative Property of Multiplication - changing the grouping when multiplying does

not affect the answer

Example:

Multiplication Property of 0- the product of 0 and any real number is 0.

Example:

Multiplication Property of 1- the product of 1 and any real number is that number.

Example:

Multiplicative Inverses- the product of any number and its multiplicative inverse

(reciprocal) is 1.

Example:

Properties of Division

Dividing by 1- the quotient of dividing by 1 is the original number.

Example:

Dividing by itself- the quotient of dividing by itself is 1.

Example:

Division with 0- the quotient of dividing by 0 is undefined. Dividing 0 by any real number (except 0) is 0.

Example:

How do you MULTIPLY/DIVIDE REAL NUMBERS?

______________________________________________________________

Section 1.5: Multiplying/Dividing Real Numbers Practice Problems

Solve the following problems

1a. _________ b. _________ c. ________

2a. __________ b. __________ c. _________

3a. __________ b. ________ c. _________

4a. __________ b. __________

5. Create a word problem where to find the solution you need to multiple or divide a negative and positive value.

______________________________________________________________________________

1.6: Exponents and Order of Operations

Exponents- used to indicate repeated multiplication.

2 is the base and 3 is the exponent.

Practice Examples:

Write each expression using exponents:

a. = __________ b. = __________

c. fourteen cubed = __________ d. = _________________

e. = __________ d. = _________________

Write out each expression and find the value:

a. = _______________________ b. = _________________________

c. = _______________________ d. = _________________________

e. = __________________ f. = _________________________

Solve:

What are the steps for ORDER OF OPERATIONS?

Order of Operations

1) Parenthesis

2) Exponents

3) Multiplication/ Division (Left to right)

4) Addition/ Subtraction (Left to right)

Practice Examples:

a. =

b. =

Practice Examples continue:

c. = d. =

e. = f. =

What are the steps for ORDER OF OPERATIONS?

______________________________________________________________________________

The Mean (Average)

Arithmetic mean (average)- divide the sum of the values by the number of values.

Example:

What is your test average if your test scores were?:

Test 1: 90

Test 2: 80

Test 3: 100

Test 4: 70

Section 1.6: Exponents/Order of Operations Practice Problems

Write each expression using exponents:

1a. = _____________ b. = __________________

2a. = __________ b. = ________________

Write out each expression and find the value:

3a. = _________________ b. = ________________________________

4a. = _________________ b. = (Don’t write out, just solve) _________

5a. = _________________ b. = _______________________________

6a. = ___________________ b. = _______________________

Solve:

7a. = ________ b. = ____________

8a. = ________ b. = ___________

9a. = ____________ b. = ___________________

10. What is the basketball point average if the points per game were?: ______________

Game 1: 14 Game 2: 18 Game 3: 20 Game 4: 28 Game 5: 21 Game 6: 32

1.7: Algebraic Expressions

What are ALGEBRAIC EXPRESSIONS and why do we use them?

Algebraic expressions- variables and/or numbers can be combined with the operations

of arithmetic

Example:

Term- part of an algebraic expression that is separated by addition or subtraction

Example: The terms of the algebraic expression are: , , and

Coefficient- the numerical factor of a term

Example: The coefficient of the terms, , and are: 3, 5, -7

Algebraic equation- equation that contains variables

Example:

Practice Example:

Given the algebraic expression:

What are the terms? _________________________________

What are the coefficients? _________________________________

Practice Example:

Given the algebraic expression:

What are the terms? _________________________________

What are the coefficients? _________________________________

Translating from Words to Symbols

Addition
Sum of x and 5	x + 5
f plus 9	f + 9
5 added to b	5 + b
8 more than r	r + 8
15 greater than g	g +15
Y increased by h	Y + h
Exceeds 6 by u	6 + u

Subtraction
Difference of 5 and h	5 - h
100 minus b	100 – b
25 less than w	w – 25
8 decreased by J	8 – J
B reduced by 7	B – 7
10 subtracted from v	v - 10
L less 4	L - 4

Multiplication
product of 5 and x	5x
25 times g	25g
twice w	2w
triple x	3x
of P	P
x squared

Division
quotient of x and 10
W divided by S
ratio of 5 to b
D split into 6 parts

Equals
X plus 5 equals 7	X + 5 = 7
Twice b results in 10	2b = 10
6 times a number is 12	6x = 12

Special
two consecutive numbers	x and (x +1)
two consecutive even (or odd) numbers	x and (x + 2)

Practice Examples: Translate into an algebraic expression or algebraic equation

1. Five times b plus twice w

________________________________________________________________________

2. The product of 4 and Y is less than 12

________________________________________________________________________

3. If 4 times a number is increased by 13, the result is 40 less than the square of the number.

________________________________________________________________________

4. The sum of a number and 9 is 5 more than twice the number.

________________________________________________________________________

5. If 7 times the sum of a number and 3 is equal to 15.

________________________________________________________________________

6. If 10 times a number is decreased by 25, the result is 12 less than twice the number.

________________________________________________________________________

7. The product of a number and the next consecutive even number is 6.

________________________________________________________________________

Evaluating Algebraic Expressions

To evaluate an algebraic expression, substitute given numbers for each variable and do the necessary calculations.

Example:

Evaluate the expression given x = 2 and y = -1.

Solution:

Substitute the values for x and y into the expression

Evaluate the problem

Practice Examples:

1. Evaluate the given expression when w = -3:

2. Evaluate the given expression when x = -3, y = 4, z = -6:

Practice Examples continue:

3. Evaluate the given expression when x = -4, y = -1:

4. Evaluate the given expression when x = -2, y = 3:

What are ALGEBRAIC EXPRESSIONS and why do we use them?

______________________________________________________________________________

Section 1.7: Algebraic Expressions Practice Problems

1. Given the algebraic expression:

a. What are the terms? ___________________________________________

b. What are the coefficients? ___________________________________________

Translate the following phrases into algebraic expressions or algebraic equations

2. The sum of a number and 5 ___________________________________________

3. The product of a number and 3 is less than 7 _______________________________

4. If 7 times a number is decreased by 2, the result is 10 less than the twice the number

________________________________________________________________________

5. If 4 times the sum of a number and 8 is equal to 40.

________________________________________________________________________

6. The sum of a number and the square of a number is equal to 5 less than twice the number.

________________________________________________________________________

7. The product of a number and the next consecutive odd number is 10.

________________________________________________________________________

8. Evaluate the given expression when: a = 6, b = -4, c = -3: ____________

9. Evaluate the given expression when x = -2: ____________

10. Evaluate the given expression when x = -3, y = 10: ____________

Chapter 2: Algebraic Expressions and Equations

2.1: Simplifying Algebraic Expressions

2.2: Solving Equations

2.3: Proportions

2.4: Formulas

2.5: Solving Inequalities

2.6: Problem Solving

2.1 Simplifying Algebraic Expressions

Why do we need to simplify algebraic expressions?

Example:

Simplify:

Solution:

Practice Examples:

Simplify

a. b.

c. d.

Distributive Property

Solve:

Example: Use distributive property to simplify

Solution:

Practice Examples:

Simplify

a. b.

c. d.

Like Terms- are terms with exactly the same variables raised to exactly the same powers.

Like Terms Unlike Terms

2x and 5x 2x and 3y

and and

Practice Examples:

Simplify

a. b.

c. d.

Why do we need to simplify algebraic expressions?

______________________________________________________________________________

Section 2.1: Simplifying Algebraic Expressions Practice Problems

Simplify

1. _______________ 2. _______________

3. _______________ 4. _______________

5. _______________ 6. _______________

7. _______________ 9. _______________

10. _______________ 11. _______________

12. _______________ 13.

____________________________________

2.2 Solving Equations

How do you solve equations?

Equation- statement indicating that two expressions are equal

Example: Is 10 a solution of: -3x + 5 = 5x – 25 ?

Solution:

10 is not a solution of: -3x + 5 = 5x – 25

Practice Examples:

a) Is -2 a solution of: ?

b) Is -1 a solution of: ?

Solving Equations (One Step)

Example: Example:

Solution: Solution:

Check: Check:

Practice Examples: Solve for x

Example: Example:

Solving Equations (Two Step)

Example: Example:

Solution: Solution:

Check: Check:

Practice Examples: Solve for x

Example: Example:

More Solving Equations

Practice Examples:

Solving Equations with variables on both sides

Example:

Solution:

Practice Examples:

a. b.

c. d.

Special Cases

e. f.

Solving Equations with fractions

Example:

Solution:

Practice Examples: Solve for x

How do you solve equations?

______________________________________________________________________________

Section 2.2: Solving Equations Practice Problems

1. Is -3 a solution of: ? 2. Is -2 a solution of: ?

Solve for the variable. Check your work.

3. _______________ 4. _______________

5. _______________ 6. _______________

7. _______________ 8. _______________

9. _______________ 10. _______________

Section 2.2: Solving Equations Practice Problems Continue

11. _______________ 12. _____________

13. _______________ 14. ____________

15. _______________ 16. _______________

17. _______________ 18. ___________

19. ____________ 20. ___________

2.3: Proportions

What are proportions and how do you solve them?

Example:

Write a proportion that solves the problem:

A man can eat 5 hamburgers in 2.5 minutes.

How many hamburgers can the man eat in 10 minutes?

a. Set up the proportion:

b. Solve the proportion:

Solution:

a. Set up the proportion:

b. Solve the proportion:

In 10 minutes the man can eat 20 hamburgers.

Practice Example:

Write a proportion that solves the problem:

A motorcycle can travel 600 miles on 20 gallons of gasoline.

How many gallons of gas are needed to travel 100 miles?

a. Set up the proportion:

b. Solve the proportion:

Write a proportion that solves the problem:

Jim can type 120 words in 3 minutes.

How many minutes would it take Jim to type 500 words?

a. Set up the proportion:

b. Solve the proportion:

Proportions

Proportion- is a mathematical statement that two equal ratios

x

10

Example:

Solve for x using proportions

Solution:

Solve proportion by multiplying the diagonals.

Practice Examples:

Solve for x using proportions

12

x

________________________________________________________________________

Example:

Solve for x using proportions

18

x

A 25 foot streetlight casts light pass a 5 foot tall person, causing a shadow. The person is standing at the base of the streetlight. How long is the shadow?

What are proportions and how do you solve them?

______________________________________________________________________________

2.3: Proportions Practice Problems

1. Solve for x using proportions

20

30

18

x

2. Solve for x and y using proportions

x

5

2.3: Proportions Practice Problems Continue

3. Set up a proportion of the following situation (Do not solve the proportion):

Jim can paint a room in 3 hours. How long would it take Jim to paint 5 rooms?

Circle which of the following proportions are the same as your proportion?

4. Set up and solve the following proportion:

A car can travel 500 miles on 25 gallons of gas.

How many miles can the car travel on 10 gallons of gas?

5. Set up and solve the following proportion:

If 5 pounds of candy cost $6, how much does 7 pounds of candy cost?

2.4 Formulas

Why do we need to solve for variables in Formulas?

Formulas used in the real world:

Retail price = cost + markup r = c + m

Interest = (Principle)(Rate)(Time)

Perimeter of rectangle = 2 (Length) + 2 (Width)

Volume of Cylinder = (Radius)(Height)

Example:

Find the perimeter of a rectangle given the length is 12 feet and the width is 10 feet.

Example:

Find the length of a rectangle given the perimeter is 100 feet and the width is 15 feet.

Solving Formulas for specific variable:

Solve for m: r = c + m

Solve for R:

Solve for h:

Solving equation for specific variable:

Example: Solve for z: x = -5u – 4z

Solution:

Step 1: Get z on one side:

x = -5u – 4z

+5u +5u

x + 5u = -4z

Step 2: Get z alone:

x + 5u = -4z

-4 -4

Step 3: Divide both terms by -4

Step 4: Pull the fraction in front:

Practice Examples:

Solve for x: w = 2x + 4z Solve for b: 6a = -3b + 4c

Why do we need to solve for variables in Formulas?

______________________________________________________________________________

Section 2.4: Formulas Practice Problems

1. The formula for the volume of a rectangle is: Volume = Length(Width) (Height) or V = lwh,

Find the height given the Volume of a rectangle is , the length is 12 feet, and the width is 5 feet.

2. The formula for the surface area of a rectangular solid is:

Surface Area = 2(length)(width) + 2(length)(height) + 2(width)(height) or SA = 2lw + 2lh + 2wh

Find the width given the Surface Area of a rectangle solid is , the length is 4 meters, and the height is 3 feet.

3. The formula to find distance given rate and time is: Distance = rate(time) or D = rt

Solve for “r”

4. The area of a trapezoid is: Area = ½ (Height)(Base 1 + Base 2) or A = ½ h (b1 + b2)

Solve for “b1”

Section 2.4: Formulas Practice Problems Continue

5. Solve for x: 3d = x – 2w 6. Solve for w: 3a = -3wz

7. Solve for b: p = 3b + 4v 8. Solve for y: 2x = -h - 5y

9. Solve for x: 8y = 8x + 4z 10. Solve for z: 7x = 2y + 10z

2.5 Solving Inequalities

What is different when you solve for inequalities?

Inequality Symbols

is less than

is greater than

is less than or equal to

is greater than or equal to

Notation when using a line graph

is used for less than or greater than

is used for less than or equal to or greater than or equal to

Notation when using interval notation

is used for less than or greater than

is used for less than or equal to or greater than or equal to

Example:

Graph

Use line graph:

Use interval notation

Example:

Graph

Use line graph:

Use interval notation

Example:

Graph or it can be written

Use line graph:

Use interval notation

Example:

Graph

Use line graph:

Use interval notation

Solving Inequalities

Example: Is 10 a solution to?

If you:

1) Multiply or Divide

2) Both sides

3) By a negative

Then the Inequality flips

Example: Solve for x:

Solution: Solve for x:

Example: Solve for x:

Solution: Solve for x:

Practice Examples:

Solve for x:

Solve for y:

Compound Inequality

Solve for x:

What is different when you solve for inequalities?

______________________________________________________________________________

Section 2.5: Solving Inequalities Practice Problems

1. Graph 2. Graph

Use line graph: Use line graph:

Use interval notation _______________ Use interval notation _______________

3. Graph 4. Graph

Use line graph: Use line graph:

Use interval notation _______________ Use interval notation _______________

Practice Examples:

5. Solve for x: 6. Solve for y:

7. Solve for x: 8. Solve for w:

Section 2.5: Solving Inequalities Practice Problems Continue

9. Solve for z: 10. Solve for x:

11. Solve for y: 12. Solve for b:

2.6 Problem Solving

How do you solve word problems?

Problem Solving Strategy

1. Analyze the problem

2. Form an equation

3. Solve the equation

4. State the conclusion

5. Check the result

Example:

A CD is priced at $15.00, but it is on sale for 20% off. What is the sale price of the CD?

1. Analyze the problem

Given: Original Price: $15.00 Discount: 20% off Unknown: x (sale price)

2. Form an equation

Original Price – Discount = Sale Price

15 - 0.20 (15) = x

3. Solve the equation

15 - 0.20 (15) = x

15 – 3 = x

x = 12

4. State the conclusion

The sale price of a CD that was $15.00 discount 20% is $12.00

5. Check the result

15 - 0.20 (15) = x

15 - 0.20 (15) = 12

15 – 3 = 12

12 = 12

Problems Solving Lab for MAT0024C

1. A pair of jeans is priced at $50.00, but is on sale for 25% off. What is the sale price of the pair of jeans?

a. $37.50

b. $40.00

c. $66.66

d. $12.50

2. If a television costs $311 after a 35% discount, what was the original cost?

a. $276.00

b. $419.85

c. $888.57

d. $478.46

3. If a DVD player costs $306 after a 30% increase in price, what was the original cost?

a. $91.80

b. $214.20

c. $437.14

d. $235.39

4. Find the amount of money now necessary to be invested at 5% simple interest to yield $200 interest in 8 years.

a. $125.00

b. $500.00

c. $320.00

d. $2500.00

5. The length of a rectangular garden is 8 more meters than its width. Its perimeter is 76 meters. Find the length of the garden.

a. 23 meters

b. 76 meters

c. 345 meters

d. 15 meters

6. The perimeter of a triangle is 30 inches. The length of the middle side is 2 inches more than the length of the smaller side and the largest side is 4 inches more than twice the length of the smallest side. Find the length of the smallest side.

a. 6 inches

b. 8 inches

c. 4 inches

d. 2 inches

7. If 8 times a number is increased by 20, the result is 26 less than the square of the number. Choose the equation that could be used to find this number, x.

8. Identify the proportion listed below that solves the problem: A car can travel 603 miles on 11 gallons of gasoline. How far can the car travel on 36 gallons?

a. c.

b. d.

9. Two cars start from the same point and travel in opposite directions. The rate of the slower car is 15 miles per hour less than the rate of the faster car. After 8 hours they are 840 miles apart. Find the speed of the cars.

a. 90 mph and 75 mph

b. 105 mph and 90 mph

c. 60 mph and 45 mph

d. 100 mph and 85

10. If a student completed this worksheet that covers word problems that are on the state competency exam to pass this class and had some difficulty. Should the student make sure that they ask questions, always go to class, take good notes, complete their homework, and seek extra help when needed?

a. Yes

b. Yes

c. Yes

d. Yes

Answers to Word Problem Lab for MAT0024C

1. A

To find the discount you multiply the original price by the percent of the discount.

So, the discount is $50.00(0.25) = $12.50.

To find the sale price you subtract the original price from the discount.

So, to find the sale price you take 50.00 – 12.50 = 37.50. The sale price is $37.50.

________________________________________________________________________

2. D Check

Original Cost – Discount = Sale Price 478.46 - .35(478.46) = 311

X - .35X = 311 478.46 – 167.46 = 311

1X - .35X = 311 311 = 311

.65X = 311

.65 .65

X = 478.46

________________________________________________________________________

3. D

Original Cost + Increase = New Price Check: 235.39 + .30(235.39) = 306

X + .30X = 306 235.39 + 70.61 = 306

1X + .30X = 306

306 = 306

1.30X = 306

1.30 1.30

X = $235.39

________________________________________________________________________

4. B

The formula to find simple interest is: Interest = Principal x Rate x Time (I = P*R*T)

The information gives would lead to: 200 = X (0.05) (8)

Then simplify: 200 = X(0.4)

Then solve for x by dividing both sides by 0.4: 200/0.04 = X(0.04)/0.04

X = $500.00, the principal (amount to be invested) is $500.00.

Check: 200 = 500(0.05)(8)

200 = 200

________________________________________________________________________

5. A

Width: W

W + 8 Length: W + 8

Perimeter: 76

W W

W + 8

Equation: W + W + 8 + W + W + 8 = 76 Width: W = 15

Combine Like Terms: 4W + 16 = 76 Length: 15 + 8 = 23

- 16 - 16

4W = 60 Check: 15 + 23 + 15 + 23 = 76

4 4 76 = 76

Solve: W = 15

________________________________________________________________________

6. A

Small: X

Middle: X + 2

Large: 2X + 4 X X + 2

Perimeter: 30

2X + 4

The equation will be: X + X + 2 + 2X + 4 = 30

Combine like terms: 4X + 6 = 30

Solve: - 6 - 6

4X = 24 Check: 6 + 6 + 2 + 2(6) + 4 = 30

4 4 6 + 8 + 16 = 30

14 + 16 = 30

X = 6 30 = 30

Small: X = 6

Middle: 6 + 2 = 8

Large: 2(6) + 4 = 16

________________________________________________________________________

7. B

Translate the problem to:

________________________________________________________________________

8. A

Note: If your proportion does not match the

proportion on the test, multiply the

diagonals to see which matches.

Matches:

________________________________________________________________________

9. C

Type	D	R	T
Fast Car	8X	X	8
Slow Car	8(X – 15)	X – 15	8
Total	840

Fill in the 4 x 4 chart with the given information.

To fill in the distance column use the formula Distance = Rate x Time

You now have the formula:

8X + 8(X – 15) = 840

Distributive Property: 8X + 8X – 120 = 840

Combine Like Terms: 16X -120 = 840

+ 120 +120

Solve for X: 16X = 960

16 16

X = 60

Substitute X into the chart and answer the question:

Type	D	R	T
Fast Car	8(60) = 480 miles	60 mph	8
Slow Car	8(60– 15) = 360 miles	60 – 15 = 45 mph	8
Total	840 miles

________________________________________________________________________

10. UMMMMMMMMMMMMMMMMM!!!!!!! The answer is YES!!!

The competency exam is very important. To pass MAT0024C Elementary Algebra you need an 80% or better on the competency exam.

Note: Use test taking strategies like eliminating the incorrect answers first, then look at the possible choices.

How do you solve word problems?

______________________________________________________________________________

Section 2.6: Problem Solving Practice Problems

1. A CD is priced at $15.00, but it is on sale for 20% off. What is the sale price of the CD?

a. $3.00

b. $10.00

c. $18.00

d. $12.00

2. If a sony play station costs $250 after a 15% discount, what was the original cost?

e. $294.12

f. $212.50

g. $287.50

h. $399.46

3. If a palm pilot costs $1300 after a 20% increase in price, what was the original cost?

a. $1625.00

b. $1083.33

e. $1560.00

f. $1040.00

4. Find the simple interest percent if you invested $1000.00 for 5 years and you received $500.00 in interest.

a. 20%

b. 50%

c. 10%

d. 40%

5. The width of a rectangular garden is 8 meters less than its length. Its perimeter is 76 meters. Find the length of the garden.

e. 23 meters

f. 76 meters

g. 345 meters

h. 15 meters

6. The perimeter of a triangle is 51 inches. The length of the middle side is 5 inches more than the length of the smaller side and the largest side is 4 inches less than three times the length of the smallest side. Find the length of the middle side.

e. 10 inches

f. 15 inches

g. 26 inches

h. 5 inches

7. If 10 times a number is decreased by 29, the result is the product of 42 and the square of the number. Choose the equation that could be used to find this number, x.

8. Identify the proportion listed below that solves the problem: A car can travel 1200 miles on 60 gallons of gasoline. How many gallons do you need to travel 100 miles?

a. c.

b. d.

9. Two shrimp boats start from the same port at the same time, but they head in opposite directions. The slower boat travels 15 knots per hour slower than the fast boat. At the end of 12 hours, they were 600 nautical miles apart. How many nautical miles had the slow boat traveled by the end of the 12-hour period?

a. 210 nautical miles

b. 17.5 nautical miles

c. 2.5 nautical miles

d. 390 nautical miles

10. If a student had some difficulty completing this worksheet that covers word problems that are on the state competency exam, that is REQUIRED to pass this class and the student wanted to do well on the competency exam, what should the student do?

a. Ask questions

b. Always go to class and take good notes

c. Complete their homework

d. Seek extra help when needed like the instructor and the tutoring center (1-131)

e. ALL THE ABOVE!!!!

Answers to Section 2.6: Problem Solving Practice Problems

1. D

To find the discount you multiply the original price by the percent of the discount.

So, the discount is $15.00(0.20) = $3.00.

To find the sale price you subtract the original price from the discount.

So, to find the sale price you take 15.00 – 3.00 = 12.00. The sale price is $12.00.

________________________________________________________________________

2. A Check

Original Cost – Discount = Sale Price 294.12 - .15(294.12) = 250

X - .15X = 250 294.12 – 44.12 = 250

1X - .15X = 250 250 = 250

.85X = 250

.85 .85

X = 294.12

________________________________________________________________________

3. B

Original Cost + Increase = New Price Check: 1083.33 + .20(1083.33) = 1300

X + .20X = 1300 1083.33 + 216.67 = 1300

1X + .20X = 1300

1300 = 1300

1.20X = 1300

1.20 1.20

X = $1083.33

________________________________________________________________________

4. C

The formula to find simple interest is: Interest = Principal x Rate x Time (I = P*R*T)

The information gives would lead to: 500 = (1000)X (5)

Then simplify: 500 = X(5000)

Then solve for x by dividing both sides by 5000: 500/5000 = X(5000)/5000

X = 0.10, the simple interest percent would be 10%.

Check: 500 = 1000(0.10)(5)

500 = 500

________________________________________________________________________

5. A

W - 8 Width: W -8

Length: W

W W

Perimeter: 76

W - 8

Equation: W + W - 8 + W + W - 8 = 76 Width: 23 – 8 = 15

Combine Like Terms: 4W - 16 = 76 Length: 23

+ 16 +16

4W = 92 Check: 15 + 23 + 15 + 23 = 76

4 4 76 = 76

Solve: W = 23

________________________________________________________________________

6. B

Small: X

Middle: X + 5

Large: 3X - 4 X X + 5

Perimeter: 51

3X - 4

The equation will be: X + X + 5 + 3X - 4 = 51

Combine like terms: 5X + 1 = 51

Solve: - 1 - 1

5X = 50 Check: 10 + 10 + 5 + 3(10) - 4 = 51

5 5 10 + 15 + 26 = 51

25 + 26 = 51

X = 10 51 = 51

Small: X = 10

Middle: 10 + 5 = 15

Large: 3(10) - 4 = 26

________________________________________________________________________

7. B

Translate the problem to:

________________________________________________________________________

8. B

Note: If your proportion does not match the

proportion on the test, multiply the

diagonals to see which matches.

________________________________________________________________________

9. A

Type	D	R	T
Fast Boat	12X	X	12
Slow Boat	12(X-15)	X	12
Total	600

Fill in the 4 x 4 chart with the given information.

To fill in the distance column use the formula Distance = Rate x Time

You now have the formula:

12X + 12(X - 15) = 600

Distributive Property: 12X + 12X -180 = 600

Combine Like Terms: 24X - 180 = 600

+180 +180

Solve for X: 24X = 780

24 24

X = 32.5

Substitute X into the chart and answer the question:

Type	D	R	T
Fast Boat	12(32.5) =390 miles	32.5 knots	12
Slow Boat	12(32.5-15) = 210 miles	32.5-15 = 17.5 knots	12
Total	600 miles

The answer from the chart is the slow boat traveled 210 miles.

________________________________________________________________________

10. UMMMMMMMMMMMMMMMMM!!!!!!! The answer is E!!!

The competency exam is very important.

To pass MAT0024C Elementary Algebra you need an 80% or better on the competency exam.

Note: Use test taking strategies like eliminating the incorrect answers first, then look at the possible choices.

MAT0024C Beginning Algebra Name: ________________

Test 1 (Chapter 1 / 2) REVIEW

Questions from Chapter 1 and 2

1. Add: (-7) + 5 + (-3) 2. Subtract: 5.8 – (-2.8)

3. Multiply: (-7)(2)(-3) 4. Solve:

5. Solve: 6. Sovle:

7. Solve: 8. Solve: 40 = -8(x - 3)

9. Solve: 10. Graph: -18 < 3x – 6 < 9

Questions on the State Exam

11. Simplify: 12. Simplify:

13. Simplify: 14. Simplify:

15. Evaluate when w = - 4: 16. Solve for r:

17. Solve for y: 18. Solve for t:

19. If 5 less than twice the square of a number, then the result is 7 more than three times a number. Choose the equation that could be used to find this number, x.

DO NOT SOLVE THE PROBLEM, JUST SET UP THE EQUATION.

20. Write a proportion that solves the problem: A motorcycle can travel 705 miles on 19 gallons of gasoline. How many gallons of gas are needed to travel 1253 miles?

DO NOT SOLVE THE PROBLEM, JUST SET UP THE PROPORTION.

21. Simplify: 22. Simplify:

23. Simplify: 24. Solve the inequality:

Word Problems on State Exam

A pair of jeans is priced at $75.00, but is on sale for 20% off. What is the sale price of the jeans?

If a digital camera costs $375 after a 25% discount, what was the original cost?

If a DVD player costs $690 after a 15% increase in price, what was the original cost?

Find the simple interest percent to yield $100 interest in 5 years when $500 is invested.

The length of a rectangular pool is 15 less than three times its width. Its perimeter is 50 meters. Find the length of the pool.

Chapter 3: Graphing

3.1: Graphing Using the Rectangular

Coordinate System

3.2: Graphing Linear Equations Using Intercepts

3.3: Graphing Linear Equations

3.4: Slope of a Line

3.5: Slope Intercept Form

3.1: Graphing Using the Rectangular

Coordinate System

Why do we need to know how to graph using

the rectangular coordinate system?

Key points:

Axes- to locate points on a plane, two perpendicular number lines

Origin- coordinate (0, 0)

Quadrants- axes divide the plane into 4 regions

x-axis- horizontal axis

y-axis- vertical axis

Ordered pair- (x-coordinate (abscissa), y-coordinate (ordinate))

Exercise: Graph and label the points

(-3, 4), (3, 4), (-4, 0), (-2, -2), (0, -2.5), (2, -2), (4, 0)

__________________________________________________________________

Exercise: Label the coordinate for each letter

A: _____________ J: _____________ N: _____________ O: _____________

R: _____________ S: _____________ T: _____________

Graph Mathematical Relationships

The time to fill a tub of water follows the table below.

Exam the relationship between the time in minutes and the water in the tub in gallons and determine what the missing values are. Then use the graph below to label the axes, label the scales, and plot the points.

Time (minutes)	Water in tub (gallons)
0	0
1	8

3	24
4	32

Looking at the plotted points, what do you notice about the relationship between the time and the amount of water in the tub?

______________________________________________________________________________

Reading Graphs

The x-axis represents time from January 3, 2007 to September 21, 2007.

The y-axis represents the average price of gas in Florida per gallon.

How much was gas in Florida on January 12, 2007? ______________________

How much was gas in Florida on March 16, 2007? ______________________

How much was gas in Florida on September 12, 2007? _______________________

When was the gas price in Florida the least amount per gallon When: _________________

and how much was it?

How much:______________

When was the gas price in Florida the most amount per gallon When: _________________

and how much was it?

How much:______________

When was the gas price $2.75 per gallon? _______________________

Why do we need to know how to graph using

the rectangular coordinate system?

______________________________________________________________________________

3.1 Graphing Using the Rectangular Coordinate System Practice Problems

1. Plot and label the following points:

(0, 0), (1, 2), (2, -4), (-3, -3), (-4, 5.5), (0, -5), (1, 0), (6, 0), (-4.5, 0)

2. Review the table and exam the relationship. Fill in the missing values and plot the points.

Label the axes and label the scales.

Year	Price of video game
2000	60
2001	50

2003	30

2005	10

What is the relationship between year and price of video game?

______________________________________________________________________________

3.1 Graphing Using the Rectangular Coordinate System Practice Problems Continue

3. Examine the graph and answer the following questions.

What is the average temperature in Florida in January? ______________________

What is the average temperature in Florida in June? ______________________

What is the average temperature in Florida in October? _______________________

What month has the lowest average temperature in Florida Month: _________________

and what is that temperature?

Temperature: ____________

What month has the highest average temperature in Florida Month: _________________

and what is that temperature?

Temperature: ____________

When is the average temperature 80 degrees? _______________________

3.2: Graphing Linear Equations Using Intercepts

What is one way to graph linear equations?

Solutions of Equations in Two Variables

Example: Determine whether each ordered pair is a solution of: 2x – y = 10

a) (2, -6) b) (10, -10)

Example: Complete the table of solutions for : y = 2x + 3

x	y	(x, y)
-2
-1
0
1
2
3

Graph the points from the example: y = 2x + 3

Example: Complete the table of solutions for : 2x – 3y = 12

x	y	(x, y)
0
	0
3
	2
-3
	-1

Graph the points from the example: 2x – 3y = 12

Graphing Linear Equations

Linear Equation- equations that can be written as ax + by = c (standard form)

Graph: y = 2x + 4

Step 1: Find at least 3 solutions using a table of values

x	y	(x, y)

Step 2: Plot the solutions and connect the coordinates

Graph using a table of values: y = 3x – 2

x	y	(x, y)

Graph using a table of values: 2x – 4y = 6

x	y	(x, y)

What is one way to graph linear equations?

______________________________________________________________________________

Section 3.2 Graphing Linear Equations Practice Problems

1. Determine whether each ordered pair is a solution of: x – 4y = 12

a) (0, -3) b) (-4, 4)

2. Complete the table of solutions for : y = -x + 5

Then plot the coordinates.

x	y	(x, y)
-2
-1
0
1
2
3

Section 3.2 Graphing Linear Equations Practice Problems Continue

3. Complete the table of solutions for : 3x – 2y = 6

Then plot the coordinates.

x	y	(x, y)
0
	0
4
	9
-3
	-4

4. Graph using a table of values: y = 2x – 1

x	y	(x, y)

Section 3.2 Graphing Linear Equations Practice Problems Continue

5. Graph using a table of values: 5x – 2y = 10

x	y	(x, y)

6. Write a linear equation for this situation and graph.

The cost to rent a car you must pay $30.00 and $0.50 per mile, where the total cost is “y” and the amount of miles used is “x”

______________________________________________________________________________

Graph the linear equation using a table of values. Make sure you label x axis, y axis, and scale.

Hint: Use realistic values for x, number of miles.

x	y	(x, y)

3.3: Graphing Linear Equations

What is another way to graph linear equations?

Linear Equation- equations that can be written as ax + by = c (standard form)

Linear Function- written as

Intercepts

x –intercept -the point where the graph crosses the x –axis (horizontal intercept)

Find the x-intercept by making y = 0

y –intercept -the point where the graph crosses the y –axis (vertical intercept)

Find the y-intercept by making x = 0

Graph the given the equation: 2x – 3y = 6

Find the x-intercept: ____________

x	y

Find the y-intercept: ____________

x	y

Graph the equation

Practice Examples

Graph the given the equation: 3x = -5y + 8

Find the x-intercept: ____________ Find the y-intercept: ____________

x	y

x	y

Graph the given the equation: y = -3x + 6

Find the x-intercept: ____________ Find the y-intercept: ____________

x	y

x	y

What is another way to graph linear equations?

______________________________________________________________________________

Section 3.3 Graphing Linear Equations Using Intercepts Practice Problems

1. Graph the given the equation: 3x – 4y = 12

Find the x-intercept: ____________ Find the y-intercept: ____________

x	y

x	y

2. Graph the given the equation: y = 4x - 5

Find the x-intercept: ____________ Find the y-intercept: ____________

x	y

x	y

Section 3.3 Graphing Linear Equations Using Intercepts Practice Problems Continue

4. Graph the given the equation: 5. Graph the given the equation:

y = 2x + 3 3x = y - 7

Find the x-intercept: ____________ Find the x-intercept: ____________

Find the y-intercept: ____________ Find the y-intercept: ____________

5. Graph the given the equation: 6. Graph the given the equation:

Find the x-intercept: ____________ Find the x-intercept: ____________

Find the y-intercept: ____________ Find the y-intercept: ____________

3.4: Slope of a Line

What is the slope of a line?

Slope- of a line is the ratio as we move from one point to another.

Example: Find the slope of this example:

Practice Examples:

Find the slope of the following graphs

m = _________ m = _________

What does slope mean?

Slope- is the rate of change.

Example:

Al would like to improve his health by joining a gym. To join a gym there is a one time membership fee of $100 and a monthly fee of $30.

Write an equation to describe this situation using “y” as the total cost and “x” as the number of months going to the gym.

The equation for this situation is:

__________________________________________________

What part of the situation is changing when x, the amount of months, changes?

Is it the one time membership fee or the total charge of the monthly fees?

__________________________________________________________________

To find the slope of this situation:

Label the variables:

What does the slope mean in this case?

__________________________________________________________________

Find the slope given two points

To find slope given two points we use the slope formula.

Slope formula:

Example: Find the slope using the points (6, -3) and (4, 3)

Step 1: Label the points

(6, -3) (4, 3)

Step 2: Substitute values into the slope formula and solve.

Answer: The slope is -3

________________________________________________________________________

Example:

Find the slope given the points (3, 6) and (5, 2)

Example:

Find the slope given the points (-2, -2) and (-12, -8)

Example:

Find the slope given the points (8, -4) and (8, -3)

Horizontal and Vertical Lines

Horizontal Lines

The graph of a horizontal line is: y = k (k is a constant)

The slope of a horizontal line is zero.

Example: Graph y = 2 Example: Graph y = -6

____________________________________________________________________________

Vertical Lines

The graph of a vertical line is: x = k (k is a constant)

The slope of a vertical line is undefined.

Example: Graph x = 5 Example: Graph x = -7

__________________________________________________________________

Practice: Graph y = -4 Practice: Graph x = 1

What is the slope of a line?

__________________________________________________________________

Section 3.4 The Slope of a Line Practice Problems

1. Given the picture find the slope: 2. Given the picture find the slope:

m = _________ m = _________

3. Given the picture find the slope: 4. Given the picture find the slope:

m = _________ m = _________

5. To join an online video game website you need to pay a one time registration fee of $20 and $1.50 per hour to use the website.

What is the equation for the cost of joining the website where “y” is the total cost and “x” is the number of hours on the website?

____________________________________________________________________________

What is the slope in this situation?

____________________________________________________________________________

Explain the slope in the context of the situation?

____________________________________________________________________________

Section 3.4 The Slope of a Line Practice Problems Continue

6. Find the slope given the points (3, 7) and (2, 10)

7. Find the slope given the points (3, -5) and (-6, -7)

8. Find the slope given the points and

9. Graph x = 3 10. Graph x = 0

11. Graph y = -4 12. Graph y = 5

3.5: Slope Intercept Form

Slope-Intercept Form: y = mx + b

m = slope b = y-intercept

_____________________________________________________________

An example of graphing using slope-intercept form: Graph:

Step 1: Determine the “m” (slope) and “b” (y –intercept)

The equation can be written as: to clearly see “m” and “b”.

The m (slope) is: The b (y-intercept) is: -1

Step 2: First plot your “b” (y-intercept) on the coordinate system. b = -1

Step 3: Use the “m” (slope) to find another coordinate. The slope is which means from the y intercept you should go down 2 and to the right 3.

Step 4: Connect the coordinates and you have graphed:

Practice Graphing Linear Equations using the Slope-Intercept Form

Graph the equation

m = _________ b = _________

Graph the equation

First step solve the equation for “y”

m = _________ b = _________

What’s My Line?

For each of the lines shown below, use your knowledge of slope and intercept to find the equation of the line given. Give your answer in the form: .

Each tick mark represents 1 unit.

1. y = 2. y =

3. y = 4. y =

5. y = 6. y =

7. y = 8. y =

9. y = 10. y =

Practice Graphing Linear Equations using the Slope-Intercept Form

Graph the equation

m = _________ b = _________

Graph the equation

First step solve the equation for “y”

m = _________ b = _________

Graphing Using the Calculator

Graph:

Step 1: Press the button on the top left of the calculator.

The screen should look like this:

Step 2: After the type in the equation.

*Be careful when you type in the fractions and negatives.

It is recommended to put in the fractions in parenthesis.

There are different buttons for subtraction sign – and negative sign (-).

Here is how you should type the equation in to the calculator: (-2/3)x-1

This is how the screen should look:

Step 3: Then press button to graph this equation.

Your graph should look like this in a standard window.

Equation of a Line using a Real Life Application “Going to the Movies”

Al is inviting his friends to go to the movies on Friday night, but he does not know how many of them are coming and what the total cost will be. He knows that he has to pay $5.00 for parking at Pointe Orlando. Also, the cost of the movies is $10.00 per person. (Movie Prices)

Write a linear equation in for the following example using “x” to represent the number of friends going to the movies and “y” representing the total cost.

________________________________________________________________________

Answer the following questions for this real life word problem

How much will it cost if Al invites 1 friend to the movies?

(Do not count Al, this is just the price for the friend) _________

After you find the solution, plot the solution (x, y) on the graph below.

How much will cost if Al invites three friends to the movies?

(Do not count Al, this is just the price for the friends) _________

After you find the solution, plot the solution (x, y) on the graph below.

If Al decided to pay for the trip out to the movies

for his friends, how many people can Al pay for with $55.00? _________

After you find the solution, plot the solution (x, y) on the graph below.

If Al decided to pay for the trip out to the movies

for his friends, how many people can Al pay for with $75.00? _________

After you find the solution, plot the solution (x, y) on the graph below.

Graph

Y-axis:

Total Cost

0        1        2        3        4        5        6       7        8       9       10

                       X-axis:   Number of Friends

100

90

80

70

60

50

40

30

20

10

0

What do you notice about the points that were plotted?

_________________________________________________________

Is there a relationship between the (x) number of friends and the (y) total cost to go to the movies? If so, what is the relationship?

Hint: Look at the cost as you add one more friend.

_________________________________________________________

What is the cost if no one goes to the movies and why?

_________________________________________________________

3.5: Slope Intercept Form Practice Problems

1. Given the equation 2. Given the equation

What is the slope (m)? ________ What is the slope (m)? ________

What is the y-intercept (b)? ________ What is the y-intercept (b)? ________

3. Given the equation 4. Given the equation

What is the slope (m)? ________ What is the slope (m)? ________

What is the y-intercept (b)? ________ What is the y-intercept (b)? ________

5. Graph using slope-intercept form 6. Graph using slope-intercept form

m = _______ b = _______ m = _______ b = _______

3.5: Slope Intercept Form Practice Problems Continue

7. Graph using slope-intercept form 8. Graph using slope-intercept form

m = _______ b = _______ m = _______ b = _______

9. Find the equation of the line in 10. Find the equation of the line in

slope-intercept form of the graph below: slope-intercept form of the graph below:

___________________________ ___________________________

3.5: Slope Intercept Form Practice Problems Continue

11. You are trying to raise money to buy a new cell phone. You already have $50.00 saved up and you are working at a job where you make $10 per hour.

a. Write an equation in slope-intercept form that explains this relationship where “x” is the number of hours you work and “y” is the total amount you have earned?

______________________________________________________________________

b. What is the slope? _________

What does the slope mean in the context of the problem?

_______________________________________________________________________

c. What is the y-intercept? _________

What does the y-intercept mean in the context of the problem?

_______________________________________________________________________

d. How much money would you have saved up after 8 hours of work? ____________

e. How much money would you have saved up after 16 hours of work? ____________

f. How many hours do you have to work if the cost of the cell phone is $300? _________

Chapter 4: Polynomials

4.1: Rules for Exponents

4.2: Zero and Negative Exponents

4.3: Scientific Notation

4.4: Polynomials

4.5: Adding/ Subtracting Polynomials

4.6: Multiplying Polynomials

4.7: Division of Polynomials (Monomials)

4.1: Rules for Exponents

2 is the base and 4 is the exponent

Example:

Simplify:

Solution:

Practice Example:

Simplify:

When multiplying exponential expressions with the same base, what is the short cut to simplify the expression?

________________________________________________________________________

Properties of Exponents

Examples:

a. b.

c. d.

Example:

Simplify:

Solution:

Practice Example:

Simplify:

When dividing exponential expressions with the same base, what is the short cut to simplify the expression?

________________________________________________________________________

Properties of Exponents

Examples:

a. b.

c. d.

Example:

Simplify:

Solution:

Practice Example:

Simplify:

When raising exponential expressions to a power, what is the short cut to simplify the expression?

________________________________________________________________________

Properties of Exponents

Examples:

a. b.

c. d.

4.1: Rules for Exponents Practice Problems

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

4.1: Rules for Exponents Practice Problems Continue

13. 14. 15.

16. 17. 18.

19. 20. 21.

22. 23.

4.2: Zero and Negative Exponents

Example:

Simplify:

One Solution:

What is a base raised to 0?

Practice Example:

Simplify:

When raising an exponential expression to zero, what is the simplified expression (except for 0 and infinity)?

________________________________________________________________________

Properties of Exponents

Examples:

a. b.

c. d.

e. f.

Example:

Simplify:

One Solution:

What do we do with a negative exponent?

So,

Practice Example:

Simplify:

How do we make a negative exponent positive?

________________________________________________________________________

Properties of Exponents

Examples:

a. b.

c. d.

e. f.

4.2: Zero and Negative Exponents Practice Problems

1. 2. 3.

4. 5. 6.

7. 8. 9.

4.2: Zero and Negative Exponents Practice Problems Continue

10. 11. 12.

13. 14. 15.

16. 17. 18.

4.3: Scientific Notation

Why do we use scientific notation?

Scientists often deal with extremely large and small numbers.

For example, the distance from the earth to the sun is 150,000,000 kilometers (93,750,000 miles) apart.

Another example is the diameter of the influenza virus is 0.00000256 inch.

Standard Form: Scientific Notation:

2,500,000

_______________________________________________________________________

Standard Form: Scientific Notation:

0.000123

________________________________________________________________________

Scientific Notation: Standard Form:

-325,000

_______________________________________________________________________

Scientific Notation: Standard Form:

0.0000005

_______________________________________________________________________

Practice Examples:

Convert to scientific notation:

a. 3,650,000,000 _______________________

b. 0.0093 _______________________

c. -0.000000004 _______________________

Convert to standard notation:

a. _______________________

b. _______________________

c. _______________________

Why do we use scientific notation?

________________________________________________________________________

4.3: Scientific Notation Practice Problems

Convert to scientific notation:

1. 2,500,000 2. -0.000004653

3. 0.0012 4. -5,000,000,000

Convert to standard notation:

5. 6.

7. 8.

4.4: Polynomials

What is a polynomial?

Polynomial- is an expression written as a sum of terms.

Ex. , , ,

Monomial- polynomial with 1 term

Ex. , , ,

Binomial- polynomial with 2 terms

Ex. , , ,

Trinomial- polynomial with 3 terms

Ex. , ,

Polynomial with _____ terms- polynomial with more than 3 terms

Ex. Polynomial with 4 terms:

Ex. Polynomial with 5 terms:

What type of Polynomial are these?

_______________________________

Degree of a term: of a polynomial in one variable is the value of the exponent on the variable. If a polynomial is in more than one variable, the degree of a term is the sum of the exponents on the variables. The degree of a nonzero constant is 0

Degree of a polynomial: the highest degree of any term of the polynomial.

Example:

Given the polynomial:

List the terms: __________________________________________

List the coefficients of each term: __________________________________________

Find the degree of each term: __________________________________________

Find the degree of the polynomial: __________________________________________

Practice Example:

Given the polynomial:

List the terms: __________________________________________

List the coefficients of each term: __________________________________________

Find the degree of each term: __________________________________________

Find the degree of the polynomial: __________________________________________

Evaluate a polynomial:

Evaluate: when x = -1

Evaluate: when x = -2

Graphing nonlinear equations:

Use the table of values

Graph:

x	y

Graph:

x	y

What is a polynomial?

________________________________________________________________________

4.4: Polynomials Practice Problems

1. What types of Polynomial are these?

Choices: monomial, binomial, trinomial, polynomial with ____ terms

a. _______________________________

b. _______________________________

c. _______________________________

d. _______________________________

e. _______________________________

2. Given the polynomial:

a. List the terms: __________________________________________

b. List the coefficients of each term: __________________________________________

c. Find the degree of each term: __________________________________________

d. Find the degree of the polynomial: __________________________________________

3. Given the polynomial:

a. List the terms: __________________________________________

b. List the coefficients of each term: __________________________________________

c. Find the degree of each term: __________________________________________

d. Find the degree of the polynomial: __________________________________________

4.4: Polynomials Practice Problems Continue

Evaluate questions 4 to 7

4. when x = -1 5. when x = -2

6. when x = -3, y = -2 7. when x = 2, y = -1

8. Graph:

x	y

4.5: Adding/ Subtracting Polynomials

How do you add and subtract polynomials?

Simplify:

Add

Solution:

Simplify:

Subtract

Solution:

Practice Problems:

a. b.

c. d.

e. f.

How do you add and subtract polynomials?

________________________________________________________________________

4.5: Adding/ Subtracting Polynomials Practice Problems

Simplify:

1. 2.

3. 4.

5. 6.

4.5: Adding/ Subtracting Polynomials Practice Problems Continue

Simplify:

7. 8.

9. 10.

11. 12.

4.6: Multiplying Polynomials

How do you multiply polynomials?

Simplify:

Solution:

Distributive Property

Practice Exercises:

Simplify:

Solution:

Method 1: FOIL

Method 2: Table Method

-9

Practice Problems:

Optional Table Method

Special Products

Examples:

Simplify:

Optional Table Method

Simplify:

Optional Table Method

Simplify:

Optional Table Method

Simplify:

Optional Table Method

How do you multiply polynomials?

______________________________________________________________________________

4.6: Multiplying Polynomials Practice Problems

Simplify:

1. 2.

3. 4.

5. 6.

7. 8.

4.6: Multiplying Polynomials Practice Problems Continue

Simplify:

9. 10.

11. 12.

13. 14.

15. 16.

4.7: Division of Polynomials (Monomials)

How do you divide polynomials (monomials)?

Example:

Simplify

Solution:

Practice Examples:

a. b.

Example:

Simplify

Solution:

Practice Examples:

a. b.

How do you divide polynomials (monomials)?

______________________________________________________________________________

4.7: Division of Polynomials (Monomials) Practice Problems

Simplify:

1. 2.

3. 4.

5. 6.

7. 8.

MAT0024C Beginning Algebra Name: ________________

Test 2 (Chapter 3 / 4) REVIEW

Questions from Chapter 3

1. Plot the following points and label 2. Is (-2, 5) a solution to:

A (-2, 3) B (0, 5) -x + y = 3

C (-6, 0) D (0, 0)

E (-7, -1)

3. Complete the table for the equation: 4. Plot the points in #3

3x – y = 6

x	y	(x, y)
-2
-1
0
	6
	-3
	3

5. Find the x-intercept of: y = -3x + 6 6. Find the y-intercept of: 2x – 3y = 12

7. Find the slope of: (-3, 2) and (-5, -4) 8. Graph:

9. What is the equation of the line for: 10. Graph:

______________________________

Questions from Chapter 4

11. Simplify: 12. Simplify:

13. Simplify: 14. Simplify:

15. Simplify: 16. Simplify:

17. Simplify: 18. Simplify:

19. Simplify: 20. Simplify:

21. Convert to scientific notation: 22. Convert to standard notation:

Review Questions on the State Exam

23. Simplify: 24. Evaluate when w = -1:

25. Solve for r: 26. Solve for y:

27. Solve for b: 28. If a digital player costs $425 after a

15% discount, what was the original cost?

29. If 8 less than the square of a number, then the result is the product of 6 and a number. Choose the equation that could be used to find this number, x.

DO NOT SOLVE THE PROBLEM, JUST SET UP THE EQUATION.

30. Write a proportion that solves the problem: A hybrid can travel 1100 miles on 55 gallons of gasoline. How many gallons of gas are needed to travel 1925 miles?

DO NOT SOLVE THE PROBLEM, JUST SET UP THE PROPORTION.

Chapter 5: Factoring

5.1: Factoring by GCF and Grouping

5.2: Factoring Trinomials Form

5.3: Factoring Trinomials Form

5.4: Factoring Difference of Two Squares

and Factor Perfect Square Trinomials

5.5: Factoring Using Multiple Methods

5.6: Solve Quadratic Equations by Factoring

5.7: Simplify Rational Expressions

5.8: Factoring Applications

5.1: Factoring by GCF and Grouping

What is factoring?

Find the prime factors of:

6 27 250

The greatest common factor (GCF)- the largest common factor of the integers.

To Find the GCF:

1. Find the prime factorization of the terms

2. Find the common factors in each of the terms and the most of each in all terms

3. Multiply the most common factors in all of the terms.

Example: Find the GCF of: 36 and 90

The common factors are: 2, 3

The most 2’s in common is: 2

The most 3’s in common is:

The GCF is:

Practice Example: Find the GCF of: 24 and 70

Practice Example:

Find the GCF of: 45 60 75

Example: Find the GCF of: and

The common factors are: 3,x, y

The most 3’s in common is:

The most x’s in common is:

The most y’s in common is:

The GCF is:

Practice Example: Find the GCF of: and

Practice Example:

Find the GCF of:

Simplify:

Now work backwards to get the original back

Factor using Greatest Common Factor

Example: Factor-

If we break down the polynomial into prime factors we would have:

What are the most factors you can pull out of all of the terms?

Then what remains once that is pulled out?

Divide by what you are factoring out to see what remains.

The factored form is:

You can check by simplifying.

Practice Examples:

Factor:

Factor using Grouping

Factor:

Example: Factor-

Method 1 will not work because you can not pull out of all of the terms.

Since there are 4 terms, we can group 2 in common and another 2 in common.

and

Now, we can pull out from each group:

and

If this is factorable, then what is the parenthesis will match.

Pull out what they both have in common and see what remains.

, this is the factored form.

You can check by simplifying.

Practice Example:

Factor:

Practice Examples:

Factor:

What is factoring?

______________________________________________________________________________

5.1: Factoring by GCF and Grouping Practice Problems

1. Find the GCF of: 2. Find the GCF of:

54, 72, 90 ,

3. Find the GCF of: 4. Find the GCF of:

, , , ,

5. Factor: 6. Factor:

7. Factor: 8. Factor:

5.1: Factoring by GCF and Grouping Practice Problems Continue

9. Factor: 10. Factor:

11. Factor: 12. Factor:

13. Factor: 14. Factor:

15. Factor: 16. Factor:

5.2: Factoring Trinomials Form

How do you factor trinomials in the form ?

Simplify:

Solution:

Notice to get the first term, it is the multiplication of the F part of FOIL

Notice to get the inside term, it is the addition of the O + I of FOIL

Notice to get the outside term, it is the multiplication of L of FOIL.

Simplify:

Example: Factor-

Pulling out the GCF or Grouping will not work because you can not pull out of all of the terms and there is not an even number of terms to group.

Our goal is to factor into two factors that are being multiplied:

To get the, the factors must be in the front (F of FOIL).

To factor this polynomial where the leading coefficient is 1:

We need to two numbers that will multiply to the last term of: + 12 (L of FOIL)

and add to the middle term of: + 7 (O + I of FOIL)

We look at the factors of +12 and find: , ,

, ,

Which pair will multiply to +12 and add to +7?

The numbers are: +3 and +4.

****Make sure your signs are correct***

The factored form is:

You can check by simplifying.

Know Your Signs

Multiply: +

Then the signs are:

+ + OR - -

Add: + Add: -

+ + - -

________________________________________________________________________

Multiply: -

Then the signs are:

+ -

Practice Examples:

Factor: Factor:

How do you factor trinomials in the form ?

______________________________________________________________________________

5.2: Factoring Trinomials Form Practice Examples

1. Factor: 2. Factor:

3. Factor: 4. Factor:

5. Factor: 6. Factor:

7. Factor: 8. Factor:

5.2: Factoring Trinomials Form Practice Examples Continue

9. Factor: 10. Factor:

11. Factor: 12. Factor:

13. Factor: 14. Factor:

15. Factor: 16. Factor:

5.3: Factoring Trinomials Form

How do you factor trinomials in the form ?

Example: Factor-

Method 1 or Method 2 or Method 3 will not work because:

You can not pull out of all of the terms (Method 1 Pull out GCF),

There is not an even number of terms to group (Method 2 Grouping), and

Since the leading coefficient is not 1, (Method 3 Trinomial a = 1)

You still need two numbers that multiply to the last term, but they will NOT add to the middle term coefficient because the leading coefficient is not 1.

There are two ways to factor this polynomial: Trial and Error and Change to grouping

First Method: Trial and Error

We know that when we factor, the factored form will be:

Step 1: We will first list the possible factors:

We must get the leading term of

The possibilities are:

Step 2: We know that it must multiply to the last term of + 5:

The possibilities with signs are: **Signs are VERY important**

Since the middle term is negative, and it must add to the middle term, then it must be:

Step 3: Then try different combinations and simply each one to try to get:

Ex.

, this is the factored form. You can check by simplifying.

Second Method: Change to grouping

Example: Factor-

In this method we want the trinomial to become a polynomial with four terms so we can perform grouping. There is a specific way to change the trinomial into a polynomial with four terms.

Step 1: Find the middle factors

Multiply the leading coefficient by the last constant term.

, we want to find factors that multiply to get this number (including the sign)

The factors must also add to the middle term coefficient, in this case -23.

The factors that multiply to +60 and add to -23 are: -20 and -3

Step 2: Rewrite the polynomial

We found the two coefficients to replace the middle term to form a polynomial with 4 terms.

The trinomial will now become:

It does not matter which number goes first, but since we are going to group, you want to place factors next to terms they have something in common with.

For example, place the “-3x” next to the “” because they have a “3x” in common and place the “-20x” next to the “5” because they have a “5” in common.

Step 3: Use Grouping

, this is the factored form. You can check by simplifying.

Practice Example:

Factor: Factor:

Factor:

How do you factor trinomials in the form ?

______________________________________________________________________________

5.3: Factoring Trinomials Form Practice Examples

1. Factor: 2. Factor:

3. Factor: 4. Factor:

5. Factor: 6. Factor:

5.3: Factoring Trinomials Form Practice Examples Continue

7. Factor: 8. Factor:

9. Factor: 10. Factor:

11. Factor: 12. Factor:

5.4: Factoring Difference of Two Squares

and Factor Perfect Square Trinomials

How do you factor difference of two squares?

Binomial (+/-) Special Case (Difference of Two Squares)

Example: Factor-

Method 1 or Method 2 or Method 3 or Method 4 will not work because you can not pull out of all of the terms, there is not an even number of terms to group, and it is not a trinomial.

This is a Special Case because:

1) There are only 2 terms

2) There is a (-) in the middle

3) You can take the square root of both terms

If all the criteria above represented, then you have difference of squares.

To factor this polynomial, you take the square root of both terms:

2x and 3

Since there is no middle term, it must have canceled each other out.

Therefore, you make one factor with (+) and another factor with (-).

, this is the factored form.

You can check by simplifying.

Practice Example:

Factor: Factor:

How do you factor difference of two squares?

______________________________________________________________________________

Factoring Perfect Square Trinomial

Example: Factor-

is a perfect square trinomial because:

The first term: is the square of x

The last term: 25 is the square of 5:

The second term: 10x is twice the product of x and 5: 2(5)(x) = 10x

A trinomial is a perfect square trinomial if:

The factored form of: is:

________________________________________________________________________

Example: Factor-

is a perfect square trinomial because:

The first term: is the square of 3x

The last term: 49 is the square of 7:

The second term: -42x is - twice the product of 3x and 7: -2(3x)(7) = -42x

A trinomial is a perfect square trinomial if:

The factored form of: is:

Practice Example:

Factor: Factor:

5.4: Factoring Difference of Two Squares and Factor Perfect Square Trinomials Practice Problems

Factor the following:

1. 2. 3.

4. 5. 6.

7. 8. 9.

5.4: Factoring Difference of Two Squares and Factor Perfect Square Trinomials Practice Problems Continue

Factor the following:

10. 11. 12.

13. 14. 15.

16. 17. 18.

5.5: Factoring Using Multiple Methods

If sometimes you need to factor using multiple methods,

what process should you follow?

Steps for Factoring Polynomial

Check each method in this order:

1) Pull out the GCF

Ex.

2) Grouping (Hint: 4 Terms or terms that can be paired)

Ex.

4) Trinomial where leading coefficient is 1

Ex.

5) Trinomial where the leading coefficient is not 1

Ex.

6) Difference of Squares

Ex.

If sometimes you need to factor using multiple methods,

what process should you follow?

__________________________________________________________________

Practice Examples:

Factor: Factor:

5.5: Factoring Using Multiple Methods Practice Problems

1. Factor: 2. Factor:

3. Factor: 4. Factor:

5. Factor: 6. Factor:

5.6: Solve Quadratic Equations by Factoring

How do you solve quadratic equations by factoring?

Example: Solve: by Factoring

In this method, MAKE SURE that the quadratic equation is equal to 0

Step 1: Factor the quadratic equation

Step 2: Set each factor equal to 0

Step 3: Solve each equation

Step 4: Check your answers

Answers are x = 0.5 and x = -3

Substitute each answer individually into the original equation:

Example: Solve the quadratic equation:

Step 1: Set the quadratic equation equal to zero

Step 2: Factor the quadratic equation

Step 3: Set each factor equal to zero

Step 4: Solve each factor

Step 5: Check the solutions

Algebraically

Practice Example: Solve: by Factoring

EXAMPLE: Solve the quadratic equation:

Step 1: Set the quadratic equation equal to zero

Step 2: Factor the quadratic equation

Step 3: Set each factor equal to zero

Step 4: Solve each factor

Step 5: Check the solutions

How do you solve quadratic equations by factoring?

______________________________________________________________________________

5.6: Solve Quadratic Equations by Factoring Practice Problems

Solve:

1. 2.

3. 4.

5. 6.

5.6: Solve Quadratic Equations by Factoring Practice Problems Continue

Solve:

7. 8.

9. 10.

11. 12.

5.7: Simplify Rational Expressions

How do you simplify rational expressions?

Example: Simplify-

Solution:

Step 1: Factor the numerator and denominator

Step 2: Simplify

Step 3: Final Answer

Practice Examples:

Simplify- Simplify-

Example: Simplify-

Solution:

Step 1: Factor the numerator and denominator

Step 2: Simplify

Step 3: Final Answer

Practice Examples:

Simplify- Simplify-

Practice Examples:

Simplify- Simplify-

How do you simplify rational expressions?

______________________________________________________________________________

5.7: Simplify Rational Expressions Practice Problems

Simplify:

1. 2.

3. 4.

5. 6.

7. 8.

5.7: Simplify Rational Expressions Practice Problems Continue

Simplify:

9. 10.

11. 12.

13. 14.

5.8: Factoring Applications

What are some examples where factoring is used in life?

Word Problem:

Find the area of a rectangle given that the length of the rectangle is 5 more than twice the width.

Solution:

2x + 5

To find the area of a rectangle use the formula for Area of a rectangle: Area = Length times Width or

In this case, the width is “x” and the length is “2x+5” so the area is:

Factoring Application 1: Factoring can be used to find unknowns.

Example:

Find the length and width of a rectangle whose area is:

(In this case, the length is the longer side and the width is the shorter side.)

Once the width and length algebraic expressions are found, find the value of the width and length when x = 5 ft. Then verify the area by substituting x = 5 ft into the area expression.

Solution:

To find the solution factor the area to find the length and width:

The width is “x + 2” feet and the length “2x + 1” feet.

2x + 1

x + 2

If x = 5 ft, then:

Practice Example:

Find the length and width of a rectangle garden whose area is:

(In this case the length is the longer side and the width is the shorter side.)

Once the width and length are found, find the value of the width and length when x = 10 ft. Then verify the area by substituting x = 10 ft into the area.

Practice Example:

Find the length and width of a square play area whose area is:

(In this case the length is the longer side and the width is the shorter side.)

Once the width and length are found, find the value of the width and length when x = 6 ft. Then verify the area by substituting x = 6 ft into the area.

Factoring Application 2: Solving real life equations

The equation of a person who dives off a 48 ft. cliff into a river with an initial velocity of 32 ft/sec is:

, where “t” is time in seconds and “h” is height in feet.

When will the diver hit the water (when height = 0)?

Solution:

To find the solution:

1) Substitute 0 in for the height (h)

2) Factor the trinomial by first factoring out the -16:

3) Continue factoring by factoring the trinomial:

4) Set each term with a variable equal to the right side of 0 and solve for t:

5) The solutions are 3 seconds and -1 seconds. Since -1 seconds is not realistic, we do not include that solution. The diver will hit the water is 3 seconds.

Practice Example:

The equation of a person who dives off a 64 ft. cliff into a river with an initial velocity of 0 ft/sec is:

, where “t” is time in seconds and “h” is height in feet.

When will the diver hit the water (when height = 0)?

Practice Example:

The equation of the path of a ball thrown in the air at a speed of 96 ft/sec is: , where “t” is time (seconds) and “h” is height in feet.

When will the ball hit the ground (when height = 0)?

What are some examples where factoring is used in life?

____________________________________________________

5.8: Factoring Applications Practice Problems

Solve:

1. Find the length and width of a rectangular room whose area is:

Draw a picture of the situation.

(In this case the length is the longer side and the width is the shorter side.)

Once the width and length are found, find the value of the width and length when x = 10 ft. Then verify the area by substituting x = 10 ft into the area.

2. Find the length and width of a square room whose area is:

Draw a picture of the situation.

(In this case the length is the longer side and the width is the shorter side.)

Once the width and length are found, find the value of the width and length when x = 4 ft. Then verify the area by substituting x = 4 ft into the area.

5.8: Factoring Applications Practice Problems Continue

Solve:

3. The equation of a person who dives off a 96 ft. cliff into a river with an initial velocity of 16 ft/sec is:

, where “t” is time in seconds and “h” is height in feet.

When will the diver hit the water (when height = 0)?

4. The equation of the path of a ball thrown in the air at a speed of 16 ft/sec is: , where “t” is time (seconds) and “h” is height in feet.

When will the ball hit the ground (when height = 0)?

MAT0024C Beginning Algebra Name: ________________

Test 3 (Chapter 5) REVIEW

Questions from Chapter 5

Factor:

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

Solve:

15. = 0 16. = 0

Solve:

17. = 0 18.

19. Simplify: 20. Simplify:

21. Two machines can complete 8 tasks every 3 days. Let t represent the number of tasks these machines can complete in a 30-day month. Write a proportion to show this example.

22. Simplify: 23. Convert to scientific notation: 0.000023

24. Solve: 25. Simplify:

26. Simplify: 27. Solve for t:

28. Find the y –intercept for: 29. Graph:

_{30. The length of a rectangular pool is 6 less than
twice the width. The perimeter of the pool is 78 feet. Find the length and
width of the pool. Label each distance correctly.}

Chapter 6: Radicals

6.1: An Introduction to Square Roots

6.2: Simplify Square Roots

6.3: Adding and Subtracting Radical Expressions

6.4: Multiplying and Dividing Radical Expressions

6.5: Solving Radical Equations

6.6: Higher Order Roots

6.1: An Introduction to Square Roots

What are square roots and when are they used?

Square Root of x– the real number that when multiplied by itself, produces x

- radical symbol

- the 36 in this example is the radicand

Example: Solve:

Solution:

because

List of perfect squares:

1 4 9 16 25 36 49 64 81 100 121 144

Practice Examples:

Estimate Square Roots

Estimate the

Solution:

The is between the which is 6 and which is 7.

So the is approximately 6.7.

Estimate the following square roots.

Square Roots of Negatives

1. The square root of a perfect square is a rational number.

2. The square root of a positive non perfect square is an irrational number.

3. The square root of a negative number is a non real number.

Square Roots of Variable Expressions

because

Since we have pair, the square root of 9 is 3.

The same is true for variables.

because

Find the pattern:

What is the pattern? ______________________________________________________

________________________________________________________________________

Practice Examples:

Pythagorean Theorem- In a right triangle, the sum of the squares of the lengths of the two shorter sides (legs) is equal to the square of the length of the longest side (hypotenuse).

c

Pythagorean Theorem:

Find the length of the missing side.

x

5

13

Distance Formula- the distance d between the points with coordinates and

Find the distance between: and

What are square roots and when are they used?

______________________________________________________________________________

6.1: An Introduction to Square Roots Practice Problems

Simplify

1. __________ 2. __________ 3. __________

4. __________ 5. __________ 6. __________

Estimate the following square roots.

7. __________ 8. __________

9. __________ 10. __________

Simplify:

11. __________ 12. __________

6.1: An Introduction to Square Roots Practice Problems Continue

Simplify:

13. __________ 14. __________

15. __________ 16. __________

Find the length of the missing side.

26

17. 18.

x

24

19. Find the distance between: 20. Find the distance between:

and and

6.2: Simplify Square Roots

How do you simplify square roots?

Simplify Square Roots

Simplify

Method 1: Factor radicand into prime factors and find pairs

Step 1: Factor radicand into prime factors

Step 2: Circle the pairs

Step 3: For every pair one number comes out of the radical

Method 2: Find perfect squares

Step 1: Determine 2 numbers that multiply to 18, where one of the numbers is a perfect square. Then separate the numbers into 2 radicals.

Step 2: Take the square root of the perfect square.

Simplify completely the following square roots

Simplify

Method 1: Factor radicand into prime factors and find pairs

Step 1: Factor radicand into prime factors

Step 2: Circle the pairs

Step 3: For every pair one number comes out of the radical

Method 2: Find perfect squares

Step 1: Separate the radicand into perfect squares and non perfect squares.

Step 2: Take the square root of the perfect squares.

Practice Examples:

Simplify

Method 1: Factor radicand into prime factors and find pairs

Step 1: Factor radicand into prime factors

Step 2: Circle the pairs

Step 3: For every pair one number comes out of the radical

Method 2: Find perfect squares

Step 1: Separate the radicand into perfect squares and non perfect squares.

Step 2: Take the square root of the perfect squares.

Practice Examples:

Simplify

Quotient Rule to Simplify Square Roots

The square root of the quotient of two numbers is equal to the quotient of their square roots.

Simplify:

How do you simplify square roots?

______________________________________________________________________________

6.2: Simplify Square Roots Practice Problems

Simplify:

1. __________ 2. __________ 3. __________

4. __________ 5. __________ 6. __________

7. __________ 8. __________ 9. __________

10. __________ 11. __________ 12. __________

6.2: Simplify Square Roots Practice Problems Continue

Simplify:

13. __________ 14. __________

15. __________ 16. __________

17. __________ 18. __________

19. __________ 20. __________

6.3: Adding and Subtracting Radical Expressions

How do you add and subtract radical expressions?

Review Problems

Simplify

Ex. Ex. Ex.

Like Radicals: Square root radicals are called like radicals when they have the same radicand.

Ex. Like Radicals Ex. Unlike Radicals

Combining Like Radicals

Ex. Ex. Ex.

Simplifying Radicals

Example:

Solution:

Step 1: Simplify the radicals

Step 2: Combine Like Terms

Practice Examples:

Ex. Ex.

How do you add and subtract radical expressions?

___________________________________________________________________________

6.3: Adding and Subtracting Radical Expressions Practice Problems

Solve:

1. 2. 3.

4. 5.

6. 7.

6.3: Add and Subtract Radical Expressions Practice Continue

8. 9.

10. 11.

12. 13.

6.4: Multiplying and Dividing Radical Expressions

How do you multiply and divide radical expressions?

Example:

Solution:

Step 1: To multiply radical expressions, multiply coefficients, multiply radicands

Step 2: If needed, then simplify radicand

Practice Examples:

Ex. Ex.

Square of a square root: , for any positive real number x

Practice Examples:

Ex. Ex. Ex.

Multiplying Radicals Expressions

Example:

Solution:

Step 1: Use distributive property

Step 2: Multiply the radicals

Step 2: Simplify the radicals

Example:

Solution:

Step 1: Use distributive property or FOIL

Step 2: Multiply the radicals

Step 2: Simplify the radicals

Practice Examples:

Ex. Ex.

Ex.

Practice Examples:

Ex. Ex.

Dividing Radicals Expressions

Practice Examples:

Ex. Ex.

Applications

Find the area of the rectangle.

How do you multiply and divide radical expressions?

______________________________________________________________________________

6.4: Multiplying and Dividing Radical Expressions Practice Problems

Simplify

1. 2.

3. 4.

5. 6.

7. 8.

6.4: Multiplying and Dividing Radical Expressions Problems Continue

9. 10.

11. 12.

13. 14.

15. Find the area of square.

6.5: Solving Radical Equations

How do you solve radical expressions?

Example: Solve for x:

Solution:

Step 1: Isolate the radical

Step 2: Remove the radical by squaring both sides

Step 3: Solve for x

Step 4: Check

Practice Examples:

Example: Solve for x:

Practice Examples:

Example: Solve for x:

Applications

Solve for x, and then find the missing side.

The perimeter of the triangle is 13 feet.

3 ft.

3 ft.

Solve for x, and then find the width and length.

The perimeter of the rectangle is 22 feet.

2 ft.

2 ft.

How do you solve radical expressions?

______________________________________________________________________________

6.5: Solving Radical Equations Practice Problems

Solve:

1. 2. 3.

4. 5. 6.

7. 8. 9.

6.5: Solving Radical Equations Practice Problems Continue

Solve:

10. 11.

12. 13.

14. Solve for x, and then find the width and length.

The perimeter of the rectangle is 18 feet.

3 ft.

3 ft.

6.6: Higher Order Roots

What are and how do you solve higher order roots?

Higher Order Roots

Square roots are not the only roots. There are many different types of roots.

Here are some examples:

Simplify the following roots:

Practice examples:

Ex. Ex.

Practice examples:

Simplify

Ex. Ex.

What are and how do you solve higher order roots?

______________________________________________________________________________

6.6: Higher Order Roots Practice Problems

Simplify:

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11.

12. 13.

MAT0024C Beginning Algebra Name: ________________

Test 4 (Chapter 6) REVIEW

Questions from Chapter 6

1. Simplify: 2. Simplify:

3. Simplify: 4. Simplify:

5. Simplify: 6. Simplify:

7. Simplify: 8. Simplify:

9. Simplify: 10. Simplify:

11. Simplify: 12. Simplify:

13. Simplify: 14. Simplify:

Review Questions

15. Factor: 16. Factor:

17. Factor: 18. Factor:

19. Factor: 20. Solve:

21. Solve: 22. Simplify:

23. Simplify: 24. Solve for b:

25. Find the x –intercept for: 26. Graph:

27. Solve: 28. Simplify:

29. Translate into algebraic equation:

The square of a number less than 4 times a number is 5 more than twice a number.

_{30. The length of a rectangular pool is 5 less than
twice the width. The perimeter of the pool is 110 feet. Find the length and
width of the pool. Label each distance correctly.}

_{31. If a bed cost $675 after a 20% discount, what was
the original cost?}

MAT0020/24C

Beginning Algebra

Competency Exam

Explained with Practice Problems and Sample Exams

Created by Al Groccia for use in MAT0024C Beginning Algebra at Valencia College

MAT0020/24C Beginning Algebra

State ALGEBRA Competency EXAM

Comp Exam Question	Algebra State Test Competency
1	Order of Operations (No Grouping / No Exponents)
2	Order of Operations (with Grouping and Exponents)
3	Absolute Value (with Addition and Subtraction)
4	Simplify Algebraic Expressions (Using the Distributive Property)
5	Evaluate an Algebraic Expression
6	Solve a Linear Equation
7	Solve a Linear Equation with Fraction Coefficient(s)
8	Solve a Literal Equation
9	Translate a Word Problem into an Algebraic Expression
10	Solve a Word Problem
11	Translate a Word Problem into a Proportion
12	Simplify Exponential Expressions (Positive Integer Exponents)
13	Simplify Exponential Expressions (Positive and Negative Integer Exp.)
14	Simplify Exponential Expressions (Positive, Neg. and Zero Integer Exp.)
15	Scientific Notation (To and From)
16	Subtraction of Polynomials: (polynomial) – (polynomial)
17	Multiply a Monomial and a Binomial: (monomial)(binomial)
18	Multiply Two Binomials: (binomial)(binomial)
19	Factoring a Polynomial: Greatest Common Factor (GCF)
20	Factoring a Polynomial: Difference of Two Squares
21	Factoring a Polynomial: By Grouping
22	Factoring a Polynomial: A Trinomial
23	Simplify a Rational Expression Reduce by Factoring
24	Solving Quadratic Equations by Factoring (Leading Coefficient is ONE, a=1)
25	Solving Quadratic Equations by Factoring (Leading Coefficient is not ONE)
26	Simplify Square Root of a Monomial
27	Simplify Square Root of a Polynomial Using the Distributive Property
28	Solving a Linear Inequality
29	Identify Intercepts of a Linear (ax + by = c)
30	Match Linear Equation to Graph (ax + by = c or y = mx + b)

Question 1 on the MAT0020/24C Beginning Algebra State Exam

Order of Operations (No Grouping / No Exponents)

1. Simplify:

A. 19 B. 34 C. 7 D. 0

Solution:

Simplify:

You need to know order of operations:

1) Parentheses

2) Exponents

3) Multiplication/ Division (which ever comes first left to right)

4) Addition / Subtraction (which ever comes first left to right)

In this problem, the first step would be the division:

Then complete the subtraction from left to right.

The solution is: 0

Practice Problems for Question 1

Order of Operations (No Grouping / No Exponents)

1. Simplify:

a. -29 b. 33 c. 31 d. -43

2. Simplify:

a. b. 27 c. 23 d.

3. Simplify:

a. 31 b. -72 c. d. 66

4. Simplify:

a. 11 b. 23 c. -1 d. -5

Solutions:

1. c 2. c 3. a 4. d

Question 2 on the MAT0020/24C Beginning Algebra State Exam

Order of Operations (with Grouping and Exponents)

2. Simplify:

A. -25 B. 90 C. D.

Solution:

Simplify:

You need to know order of operations:

1) Parentheses

2) Exponents

3) Multiplication/ Division (which ever comes first left to right)

4) Addition / Subtraction (which ever comes first left to right)

In this problem, the first step would be the parenthesis:

The next step would be exponents:

The next step would be division because it occurs first from left to right:

The next step would be multiplication:

The next step would be subtraction:

The solution is: -25

Practice Problems for Question 2

Order of Operations (with Grouping and Exponents)

1. Simplify:

a. 224 b. 35 c. d. 80

2. Simplify:

a. 4 b. 20 c. 80 d. 2

3. Simplify:

a. -33 b. 66 c. -44 d. 206

4. Simplify:

a. 27 b. 51 c. 87 d. 3

Solutions:

1. d 2. b 3. c 4. a

Question 3 on the MAT0020/24C Beginning Algebra State Exam

Absolute Value (with Addition and Subtraction)

3. Simplify:

A. 16 B. 31 C. 3 D. 15

Solution:

Simplify:

You need to know order of operations.

In this problem, the first step would be to complete what is inside the absolute value, the absolute value is acting similar to parenthesis.

The next step is to take the absolute value:

The next step is complete the addition:

The solution is: 15

Practice Problems for Question 3

Absolute Value (with Addition and Subtraction)

1. Simplify:

a. 37 b. 19 c. -11 d. 7

2. Simplify:

a. 15 b. 3 c. -3 d. -15

3. Simplify:

a. 25 b. 15 c. -15 d. 3

4. Simplify:

a. 15 b. 13 c. 17 d. 5

Solutions:

1. b 2. c 3. b 4. d

Question 4 on the MAT0020/24C Beginning Algebra State Exam

Simplify Algebraic Expressions (Using the Distributive Property)

4. Simplify:

A. B. C. D.

Solution:

Simplify:

In this problem, the first step would be to simplify within the brackets. Within the brackets, use the distributive property first.

The next step is to combine like terms with the brackets.

The next step is to use distributive property.

The solution is: -42x-180

Practice Problems for Question 4

Simplify Algebraic Expressions (Using the Distributive Property)

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b. c. d.

Solutions:

1. a 2. c 3. c 4. b

Question 5 on the MAT0020/24C Beginning Algebra State Exam

Evaluate an Algebraic Expression

5. Evaluate the given expression when w = - 4:

A. – 79 B. – 119 C. – 113 D. – 73

Solution:

Evaluate the given expression when w = - 4:

In this problem, the first step would be to substitute the value into the variable.

Then using order of operations, the next step would be to complete the exponent.

The next step would be to complete multiplication from left to right.

The next step is to complete subtraction then addition left to right.

The solution is: -113

Practice Problems for Question 5

Evaluate an Algebraic Expression

1. Evaluate the given expression when w = -4:

a. 35 b. 61 c. 51 d. 45

2. Evaluate the given expression when w = -3:

a. -39 b. -27 c. -45 d. -33

3. Evaluate the given expression when x = -9, y = 4, z = -7:

a. -137 b. 151 c. 137 d. -151

4. Evaluate the given expression when x = -9, y = 2, z = -2:

a. -158 b. 158 c. -160 d. -166

Solutions:

1. c 2. a 3. a 4. d

Question 6 on the MAT0020/24C Beginning Algebra State Exam

Solve a Linear Equation

6. Solve for r:

A. B. C. D.

Solution:

Solve for r:

The first step would be to simplify both sides of the equation using distributive property.

The next step would be to move the variables to the same side of the equation.

The next step is to move the variables to other side of the equation.

The next step is to solve for r and simplify solution.

The solution is:

Note: You can check your solution by substituting it back into your equation.

Practice Problems for Question 6

Solve a Linear Equation

1. Solve for r:

a. b. c. d.

2. Solve for x:

a. b. c. d.

3. Solve for y:

a. b. c. d.

4. Solve for y:

a. b. c. d.

Solutions:

1. d 2. a 3. d 4. c

Question 7 on the MAT0020/24C Beginning Algebra State Exam

Solve a Linear Equation with Fraction Coefficient(s)

7. Solve for y:

A. B. C. D.

Solution:

Solve for y:

One way to solve this problem is to eliminate the fractions by multiply the entire equation by the least common denominator of the fractions. In this case the least common denominator is 12.

The next step is to solve for y.

The solution is:

Note: You can check your solution by substituting it back into your equation.

Practice Problems for Question 7

Solve a Linear Equation with Fraction Coefficient(s)

1. Solve for x:

a. b. c. d.

2. Solve for t:

a. b. c. d.

3. Solve for r:

a. b. c. d.

4. Solve for r:

a. b. c. d.

Solutions:

1. b 2. b 3. c 4. b

Question 8 on the MAT0020/24C Beginning Algebra State Exam

Solve a Literal Equation

8. Solve for t:

A. B. C. D.

Solution:

Solve for t:

The first step to solve for t is to move the -8z:

The next step is isolate t by dividing by 7:

Since this solution is not one of the choices we can simplify further by dividing each part of the numerator by 7.

The solution is:

Practice Problems for Question 8

Solve a Literal Equation

1. Solve for x:

a. b. c. d.

2. Solve for W:

a. b. c. d.

3. Solve for y:

a. b. c. d.

4. Solve for v:

a. b. c. d.

Solutions:

1. a 2. b 3. d 4. c

Question 9 on the MAT0020/24C Beginning Algebra State Exam

Translate a Word Problem into an Algebraic Expression

9. The sum of a number and 16 is 4 less than twice the number.

Find the equation that could be used to find this number, x.

A. B. C. D.

Solution:

The sum of a number and 16 is 4 more than twice the number.

Translating each part:

The sum of a number and 16 means: x + 16

The word “is” means “=”

Then 4 less than twice the number means: 2x - 4

The solution is: x + 16 = 2x - 4

Practice Problems for Question 9

Translate a Word Problem into an Algebraic Expression

1. If 9 times a number is increased by 20, the result is 22 less than the square of the number. Choose the equation that could be used to find this number x.

a. b.

c. d.

2. The sum of a number and 2 is 3 more than twice the number.

Find the equation that could be used to find this number, x.

a. b. c. d.

3. If 3 times the sum of a number and 5 is equal to 7.

Find the equation that could be used to find this number, x.

a. b. c. d.

4. If 6 times a number is decreased by 8, the result is 5 less than twice the number. Choose the equation that could be used to find this number x.

a. b.

c. d.

Solutions:

1. d 2. d 3. a 4. c

Question 10 on the MAT0020/24C Beginning Algebra State Exam

Solve a Word Problem

10. The length of a rectangle is 2 feet more than the width. The perimeter of the rectangle is 72 feet. Find the length.

A. 19 feet B. 17 feet C. 37 feet D. 35 feet

Solution:

The first step is to translate the statements into algebraic expressions.

The length of a rectangle is 2 feet more than the width would be represented by: x + 2

The width would be represented by: x

Here is a visual representation:

x+2

x

x

x+2

The next step is to create an algebraic equation to solve for x.

Since the perimeter is the sum of the sides, the equation would be:

(x) + (x + 2) + (x) + (x + 2) = 72

The next step is to combine like terms:

4x + 4 = 72

Then solve for x:

x = 17

Since x is the width, then the width is 17. The length is x + 2, so the length is 19.

The answer is 19.

Note: You can check your solution by substituting it back into your algebraic equation.

Practice Problems for Question 10

Solve a Word Problem

1. A CD is priced at $15.00, but it is on sale for 20% off. What is the sale price of the CD?

a. $3.00 b. $10.00 c. $18.00 d. $12.00

2. If a sony play station costs $250 after a 15% discount, what was the original cost?

a. $294.12 b. $212.50 c. $287.50 d. $399.46

3. If a palm pilot costs $1300 after a 20% increase in price, what was the original cost?

a. $1625.00 b. $1083.33 c. $1560.00 d. $1040.00

4. Find the simple interest percent if you invested $1000.00 for 5 years and you received $500.00 in interest.

a. 20% b. 50% c. 10% d. 40%

Solutions explained in following pages.

5. The width of a rectangular garden is 8 meters less than its length. Its perimeter is 76 meters. Find the length of the garden.

a. 23 meters b. 76 meters c. 345 meters d. 15 meters

a. 10 inches b. 15 inches c. 26 inches d. 5 inches

7. Two shrimp boats start from the same port at the same time, but they head in opposite directions. The slower boat travels 15 knots per hour slower than the fast boat. At the end of 12 hours, they were 600 nautical miles apart. How many nautical miles had the slow boat traveled by the end of the 12-hour period?

a. 210 nautical miles b. 17.5 nautical miles

c. 2.5 nautical miles c. 390 nautical miles

Solutions explained in following pages.

Solutions to Practice Word Problems

1. D

To find the discount you multiply the original price by the percent of the discount.

So, the discount is $15.00(0.20) = $3.00.

To find the sale price you subtract the original price from the discount.

So, to find the sale price you take 15.00 – 3.00 = 12.00. The sale price is $12.00.

________________________________________________________________________

2. A Check

Original Cost – Discount = Sale Price 294.12 - .15(294.12) = 250

X - .15X = 250 294.12 – 44.12 = 250

1X - .15X = 250 250 = 250

.85X = 250

.85 .85

X = 294.12

________________________________________________________________________

3. B

Original Cost + Increase = New Price Check: 1083.33 + .20(1083.33) = 1300

X + .20X = 1300 1083.33 + 216.67 = 1300

1X + .20X = 1300

1300 = 1300

1.20X = 1300

1.20 1.20

X = $1083.33

________________________________________________________________________

4. C

The formula to find simple interest is: Interest = Principal x Rate x Time (I = P*R*T)

The information gives would lead to: 500 = (1000)X (5)

Then simplify: 500 = X(5000)

Then solve for x by dividing both sides by 5000: 500/5000 = X(5000)/5000

X = 0.10, the simple interest percent would be 10%.

Check: 500 = 1000(0.10)(5)

500 = 500

________________________________________________________________________

5. A

W - 8 Width: W -8

Length: W

W W

Perimeter: 76

W - 8

Equation: W + W - 8 + W + W - 8 = 76 Width: 23 – 8 = 15

Combine Like Terms: 4W - 16 = 76 Length: 23

+ 16 +16

4W = 92 Check: 15 + 23 + 15 + 23 = 76

4 4 76 = 76

Solve: W = 23

________________________________________________________________________

6. B

Small: X

Middle: X + 5

Large: 3X - 4 X X + 5

Perimeter: 51

3X - 4

The equation will be: X + X + 5 + 3X - 4 = 51

Combine like terms: 5X + 1 = 51

Solve: - 1 - 1

5X = 50 Check: 10 + 10 + 5 + 3(10) - 4 = 51

5 5 10 + 15 + 26 = 51

25 + 26 = 51

X = 10 51 = 51

Small: X = 10

Middle: 10 + 5 = 15

Large: 3(10) - 4 = 26

________________________________________________________________________

7. A

Type	D	R	T
Fast Boat	12X	X	12
Slow Boat	12(X-15)	X	12
Total	600

Fill in the 4 x 4 chart with the given information.

To fill in the distance column use the formula Distance = Rate x Time

You now have the formula:

12X + 12(X - 15) = 600

Distributive Property: 12X + 12X -180 = 600

Combine Like Terms: 24X - 180 = 600

+180 +180

Solve for X: 24X = 780

24 24

X = 32.5

Substitute X into the chart and answer the question:

Type	D	R	T
Fast Boat	12(32.5) =390 miles	32.5 knots	12
Slow Boat	12(32.5-15) = 210 miles	32.5-15 = 17.5 knots	12
Total	600 miles

The answer from the chart is the slow boat traveled 210 miles.

Question 11 on the MAT0020/24C Beginning Algebra State Exam

Translate a Word Problem into a Proportion

11. Identify the proportion listed below that solves this problem:

A car can travel 500 miles on 5 gallons of gasoline.

How far can the car travel on 32 gallons of gasoline?

A. B. C. D.

Solution:

Identify the proportion listed below that solves this problem:

A car can travel 500 miles on 5 gallons of gasoline.

How far can the car travel on 32 gallons of gasoline?

The first step is to determine how to step up the way the proportion will be written.

One way is:

So the proportion set up will be:

Since that is not an option, another way to set up the proportion is:

The solution is:

Remember to solve a proportion, multiply the diagonal values.

The first step to solve: is

This is the same as: is

Practice Problems for Question 11

Translate a Word Problem into a Proportion

1. Identify the proportion listed below that solves this problem:

A car can travel 415 miles on 8 gallons of gasoline. How far can the car travel on 25 gallons?

a. b. c. d.

2. Identify the proportion listed below that solves this problem:

If 46 pounds of jelly beans cost 65 cents, how many pounds of jelly beans can be purchased for 130 cents?

a. b. c. d.

3. Identify the proportion listed below that solves this problem:

If Jim can eat 10 pies in 20 minutes, how many pies could Jim eat in 45 minutes?

a. b. c. d.

4. Identify the proportion listed below that solves this problem:

A shoe factory can produce 1000 pairs of shoes every 3 hours. How long would it take the shoe factory to produce 5000 pairs of shoes?

a. b. c. d.

Solutions:

1. a 2. b 3. a 4. d

Question 12 on the MAT0020/24C Beginning Algebra State Exam

Simplify Exponential Expressions (Positive Integer Exponents)

12. Simplify:

A. B. C. D.

Solution:

Simplify:

Know the exponent rules:

Exponent Rules:

The first step using order of operations is to complete the exponents using exponent rules.

The next step is to combine like terms using exponent rules.

The solution is:

Practice Problems for Question 12

Simplify Exponential Expressions (Positive Integer Exponents)

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b. c. d.

Solutions:

1. c 2. c 3. b 4. a

Question 13 on the MAT0020/24C Beginning Algebra State Exam

Simplify Exponential Expressions (Positive and Negative Integer Exp.)

13. Simplify:

A. B. C. D.

Solution:

Simplify:

Know the exponent rules:

Exponent Rules:

One way to simplify is to make all of the negative exponents positive by moving them from numerator to denominator or vice versa.

The next step is the combine like terms using the exponent rules.

The solution is:

Practice Problems for Question 13

Simplify Exponential Expressions (Positive and Negative Integer Exp.)

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b. c. d.

Solutions:

1. a 2. c 3. a 4. b

Question 14 on the MAT0020/24C Beginning Algebra State Exam

Simplify Exponential Expressions (Positive, Neg. and Zero Integer Exp.)

14. Simplify:

A. B. C. D.

Solution:

Simplify:

Know your exponent rules.

Exponent Rules:

The first step is to simplify the 0 exponent.

The next step is to simplify using the exponent rules.

The answer is:

Practice Problems for Question 14

Simplify Exponential Expressions (Positive, Neg. and Zero Integer Exp.)

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b. c. d.

Solutions:

1. c 2. a 3. d 4. c

Question 15 on the MAT0020/24C Beginning Algebra State Exam

Scientific Notation (To and From)

15. Convert to scientific notation: 0.000000195

A. B. C. D.

Solution:

Convert to scientific notation: 0.000000195

The first step will be to determine where the decimal place should be to convert this number from standard notation to scientific notation. In scientific notation the decimal should be after the first non zero from the left.

In this case the decimal should be after the 1 which is 7 movements to the right.

If the decimal moved to the right the exponent is negative, to the left the exponent is positive.

Then place in this form:

The solution is:

Note: This can be checked by converting from scientific notation to standard notation.

To convert to standard notation, start at the decimal point and the -7 means move the decimal point 7 movements to the left (due to the negative). A positive exponent means move the decimal to the right, a negative means move to the left.

Practice Problems for Question 15

Scientific Notation (To and From)

1. Convert to standard form:

a. b. c. d.

2. Convert to standard form:

a. b. c. d.

3. Convert to scientific notation:

a. b. c. d.

4. Convert to scientific notation:

a. b. c. d.

Solutions:

1. b 2. d 3. a 4. a

Question 16 on the MAT0020/24C Beginning Algebra State Exam

Addition/Subtraction of Polynomials

16. Simplify:

A. B. C. D.

Solution:

Simplify:

One way to simplify is to rewrite this problem by distributing the subtraction sign:

The next step is to combine like terms.

The solution is:

Practice Problems for Question 16

Addition/Subtraction of Polynomials

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b. c. d.

Solutions:

1. a 2. d 3. c 4. a

Question 17 on the MAT0020/24C Beginning Algebra State Exam

Multiply a Monomial and a Binomial

17. Simplify:

A. B. C. D.

Solution:

Simplify:

The first step to simplify is to use the distributive property and simplify.

The solution is:

Practice Problems for Question 17

Multiply a Monomial and a Binomial

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b.

c. d.

Solutions:

1. d 2. a 3. c 4. a

Question 18 on the MAT0020/24C Beginning Algebra State Exam

Multiply Two Binomials

18. Simplify:

A. B. C. D.

Solution:

Simplify:

One way to simplify this problem is with the FOIL method:

First: (4x)(6x):

Outer: (4x)(-7): -28x

Inner: (-7)(6x): -42x

Last: (-7)(-7): 49

The result is:

Then combine like terms: -28x – 42x = -70x

The solution is:

Practice Problems for Question 18

Multiply Two Binomials

1. Simplify:

a. b. c. d.

2. Simplify:

a. b.

c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b.

c. d.

Solutions:

1. c 2. a 3. d 4. a

Question 19 on the MAT0020/24C Beginning Algebra State Exam

Factoring a Polynomial: Greatest Common Factor (GCF)

19. Factor completely:

A. B.

C. D.

Solution: Factor completely:

The first step to factor this problem is to identify which method can be used to factor. In this case the factoring method is factoring by pulling out the Greatest Common Factor.

To determine the GCF of the coefficients we look at the prime factors of the coefficients:

The Greatest Common Factor of the coefficients is: 2

To determine the Greatest Common Factor of the variables, we look for the most of each variable in each term. All of the terms must have the variable for it to be factored out. It will be the smallest exponent as long as all terms have the variable.

The Greatest Common Factor for the variable y is:

The Greatest Common Factor for the variable s is:

The Greatest Common Factor is:

If we factor out from each term, we need to show what is remaining.

If we divide each term by , that will give the remaining.

The solution is:

Note: This solution can be checked by using the distributive property but be sure to

pull out the greatest common factor.

Practice Problems for Question 19

Factoring a Polynomial: Greatest Common Factor (GCF)

1. Factor:

a. b.

c. d.

2. Factor:

a. b.

c. d.

3. Factor:

a. b.

c. d.

4. Factor:

a. b.

c. d.

Solutions:

1. b 2. b 3. d 4. b

Question 20 on the MAT0020/24C Beginning Algebra State Exam

Factoring a Polynomial: Difference of Two Squares

20. Factor completely:

A. B.

C. D.

Solution: Factor completely:

The first step to factor this problem is to identify which method can be used to factor. In this case the factoring method is difference of two square because there are only 2 terms, there is a subtraction sign in the middle, and the front and back terms can be square rooted.

Make sure the expression can be factored by pulling out the Greatest Common Factor first. If you can factor by pulling out the GCF, that should be completed first.

The first step is to take the square root of the front term is:

Then take the square root of the back term is:

The solution is:

Note: This solution can be checked by simplifying through FOIL.

Practice Problems for Question 20

Factoring a Polynomial: Difference of Two Squares

1. Factor:

a. b. c. d.

2. Factor:

a. b. c. d.

3. Factor:

a. b. c. d.

4. Factor:

a. b.

c. d.

Solutions:

1. a 2. d 3. c 4. b

Question 21 on the MAT0020/24C Beginning Algebra State Exam

Factoring a Polynomial: By Grouping

21. Factor completely:

A. B. C. D.

Solution: Factor completely:

The first step to factor this problem is to identify which method can be used to factor. In this case the factoring method is grouping because there are 4 terms, the Greatest Common Factor can not be factored out of all 4 terms, but we can group terms and then factor out a Greatest Common Factor.

The next step is to group terms that have similar factors.

The next step is to factor out similar terms from each group.

The next step is to notice that both groups now have a common factor of (r + k).

Factor out the (r + k) from both group and show what remains.

The solution is:

Note: This solution can be checked by simplifying through FOIL.

Practice Problems for Question 21

Factoring a Polynomial: By Grouping

1. Factor:

a. b. c. d.

2. Factor:

a. b. c. d.

3. Factor:

a. b. c. d.

4. Factor:

a. b. c. d.

Solutions:

1. a 2. c 3. c 4. a

Question 22 on the MAT0020/24C Beginning Algebra State Exam

Factoring a Polynomial: A Trinomial

22. Identify a factor of the following trinomial:

A. B. C. D.

Solution: Identify a factor of the following trinomial:

The first step to factor this problem is to identify which method can be used to factor. In this case the factoring method is factoring a trinomial because a trinomial is given.

Make sure another factoring method such as factor out the Greatest Common factor can be completed first. If it can that should be completed first.

There are two ways to factor this polynomial: Trial and Error and Change to grouping

First Method: Trial and Error

We know that when we factor, the factored form will be:

We will first list the possible factors:

We must get the leading term of

The possibilities are:

We know that it must multiply to the last term of + 4.

The possibilities with signs are:

**Signs are VERY important**

Since the middle term is negative, and it must add to the middle term, then it must be either: or

Then try different combinations and simply each one to try get :

Ex.

The solution is:

Since the question only requires one of the factors, the solution is:

Note: This solution can be checked by simplifying through FOIL.

Second Method: Change to grouping

Solution: Identify a factor of the following trinomial:

In this method we want the trinomial to become a polynomial with four terms so we can perform grouping. There is a specific way to change the trinomial into a polynomial with four terms.

Step 1: Find the middle factors

Multiply the leading coefficient by the last constant term.

, we want to find factors that multiply to get this number (including the sign)

The factors must also add to the middle term coefficient, in this case -25.

The factors that multiply to +24 and add to -25 are: -24 and -1

Step 2: Rewrite the polynomial

We found the two coefficients to replace the middle term to form a polynomial with 4 terms.

The trinomial will now become:

It does not matter which number goes first, but since we are going to group you want to place factors next to terms they have something in common with. For example, place the “-24t” next to the “” because they have a “6t” in common.

Step 3: Use Grouping

The solution is:

Since the question only requires one of the factors, the solution is:

Note: This solution can be checked by simplifying through FOIL.

Practice Problems for Question 22

Factoring a Polynomial: A Trinomial (mixed)

1. Factor:

a. b. c. d.

2. Factor:

a. b. c. d.

3. Factor:

a. b. c. d.

4. Factor:

a. b. c. d.

Solutions:

1. a 2. d 3. b 4. c

Practice Problems for Question 22

Factoring a Polynomial: A Trinomial

1. Identify a factor of the following trinomial:

a. b. c. d.

2. Identify a factor of the following trinomial:

a. b. c. d.

3. Identify a factor of the following trinomial:

a. b. c. d.

4. Identify a factor of the following trinomial:

a. b. c. d.

Solutions:

1. d 2. a 3. c 4. a

Question 23 on the MAT0020/24C Beginning Algebra State Exam

Simplify a Rational Expression: Reduce by Factoring

23. Simplify:

A. B. C. D.

Solution:

Simplify:

The first step to simplify this expression is if possible factor the numerator and denominator.

The next step is to simplify by looking for factors that appear in the numerator and denominator that will simplify to 1.

The solution is:

Practice Problems for Question 23

Simplify a Rational Expression: Reduce by Factoring

1. Simplify:

a. b. c. d.

2. Simplify:

a. b. c. d.

3. Simplify:

a. b. c. d.

4. Simplify:

a. b. c. d.

Solutions:

1. b 2. a 3. c 4. c

Question 24 on the MAT0020/24C Beginning Algebra State Exam

Solving Quadratic Equations by Factoring

(Leading Coefficient is ONE, a=1)

24. Solve:

A. B. C. D.

Solution:

Solve:

The first step is to factor the level side of the equation.

The next step is to set each factor equal to 0.

The next step is to solve each equation.

The solutions are:

Note: The solutions can be checked by substituting them back into the original equation.

Practice Problems for Question 24

Solving Quadratic Equations by Factoring

(Leading Coefficient is ONE, a=1)

1. Solve:

a. b. c. d.

2. Solve:

a. b. c. d.

3. Solve:

a. b. c. d.

4. Solve:

a. b. c. d.

Solutions:

1. c 2. c 3. a 4. c

Question 25 on the MAT0020/24C Beginning Algebra State Exam

Solving Quadratic Equations by Factoring

(Leading Coefficient is not ONE)

25. Solve:

A. B. C. D.

Solution: Solve:

The first step is to factor the level side of the equation.

The next step is to set each factor equal to 0.

The next step is to solve each equation.

The solutions are:

Note: The solutions can be checked by substituting them back into the original equation.

Practice Problems for Question 25

Solving Quadratic Equations by Factoring

(Leading Coefficient is not ONE)

1. Solve:

a. b. c. d.

2. Solve:

a. b. c. d.

3. Solve:

a. b. c. d.

4. Solve:

a. b. c. d.

Solutions:

1. c 2. b 3. b 4. d

Question 26 on the MAT0020/24C Beginning Algebra State Exam

Simplify Square Root of a Monomial

26. Simplify completely:

A. B. C. D.

Solution:

Simplify completely:

One method of simplifying this radical expression is to break the radicand into prime factors.

The next step is for every pair, one value is brought out the radical, then simplify.

Another method is to create two radicals were one radical contains what can be square rooted (perfect squares) and the other radical contains what can not be square rooted (non perfect squares).

The next step is to take the square root of the perfect squares.

The solution is:

Practice Problems for Question 26

Simplify Square Root of a Monomial

1. Simplify completely:

a. b. c. d.

2. Simplify completely:

a. b. c. d.

3. Simplify completely:

a. b. c. d.

4. Simplify completely:

a. b. c. d.

Solutions:

1. a 2. c 3. b 4. d

Question 27 on the MAT0020/24C Beginning Algebra State Exam

Simplify Square Root of a Polynomial Using the Distributive Property

27. Simplify:

A. B. C. D.

Solution:

Simplify:

The first step to simplify this problem is to use the distributive property:

The next step is to multiply the radical expressions. Remember to multiply radical expressions, multiply the coefficients and multiply the radicands.

The next step is to simply, if needed, the radical terms.

The solution is:

Practice Problems for Question 27

Simplify Square Root of a Polynomial Using the Distributive Property

1. Simplify completely:

a. b. c. d.

2. Simplify completely:

a. b. c. d.

3. Simplify completely:

a. b. c. d.

4. Simplify completely:

a. b. c. d.

Solutions:

1. d 2. a 3. c 4. a

Question 28 on the MAT0020/24C Beginning Algebra State Exam

Solving a Linear Inequality

28. Solve the inequality:

A. B. C. D.

Solution: Solve the inequality:

The first step is to get the variables on the same side. Since this is an inequality it is preferred to have the variables on the left side because inequalities are generally written with the variable on the left so they are easier to read.

The next step is to solve for x. Remember if both sides of an inequality is multiplied or divided by a negative, the inequality changes from less than to greater than or from greater than to less than.

The solution is:

Note: This solution can be checked if you substitute a value that is such as 0 into the original equation.

Practice Problems for Question 28

Solving a Linear Inequality

1. Solve the inequality:

a. b. c. d.

2. Solve the inequality:

a. b. c. d.

3. Solve the inequality:

a. b. c. d.

4. Solve the inequality:

a. b. c. d.

Solutions:

1. c 2. d 3. c 4. b

Question 29 on the MAT0020/24C Beginning Algebra State Exam

Identify Intercepts of a Linear (ax + by = c)

29. Find the y –intercept for:

A. B. C. D.

Solution:

Find the y –intercept for:

To find the x-intercept of a linear equation make y = 0 and solve for x. To find the y-intercept of a linear equation make x = 0 and solve for y.

The first step to find the y-intercept is to substitute 0 for x and solve for y.

x	y
0	?

x	y
0	-1

The second step is to write the solution as a coordinate (x, y). The value for x is 0 and the value for y is -1.

The solution is: (0, -1)

Practice Problems for Question 29

Identify Intercepts of a Linear (ax + by = c)

1. Find the x-intercept for:

a. b. c. d.

2. Find the y-intercept for:

a. b. c. d.

3. Find the y-intercept for:

a. b. c. d.

4. Find the x-intercept and y-intercept for:

a. x-intercept: y-intercept: b. x-intercept: y-intercept:

c. x-intercept: y-intercept: d. x-intercept: y-intercept:

Solutions:

1. c 2. a 3. c 4. b

Question 30 on the MAT0020/24C Beginning Algebra State Exam

Match Linear Equation to Graph

30. Find the graph that best matches the linear equation:

A. B.

C. D.

One method is to create a table of values. Choose a value for x or y, then solve for the other variable. Three points are sufficient for a linear equation.

x	y
0	?
2	?
4	?

x	y	(x, y)
0	-4	(0, -4)
2	-3	(2, -3)
4	-2	(4, -2)

Label the points and draw a line through the points.

Another method is to use the slope intercept form (y = mx + b) to graph the equation.

The first step is to make sure that the linear equation is written as y =.

The next step is to determine the slope and y-intercept of the linear equation.

In the form: y = mx + b, the m is the slope and the b is the y intercept.

Slope (m) = and y intercept (b) = -4

To graph, first place a point on the y intercept at -4. The y intercept is the point where the graph crosses the y axis. In this case the y intercept is -4.

The next step is to use the slope to find the next point. The slope can be expressed as . The numerator determines if you move up (positive value) or down (negative value). The denominator determines if you move left (negative value) or right (positive value).