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Table of Contents. Key Terms. Pre-Assessment. Graphing/Writing with One Variable. Simple Inequalities Addition/Subtraction. Simple Inequalities Multiplication/Division. Two-Step & Multiple-Step. Compound Inequalities. Special Cases of Compound Inequalities. Key Terms.

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Presentation Transcript
slide1

Table of Contents

Key Terms

Pre-Assessment

Graphing/Writing with One Variable

Simple Inequalities Addition/Subtraction

Simple Inequalities Multiplication/Division

Two-Step & Multiple-Step

Compound Inequalities

Special Cases of Compound Inequalities

slide2

Key Terms

Return to 

Table of 

Contents

slide3

Key Terms

Equation

Inequality

Solution Set

Compound Inequality

Set Builder Notation

slide4

Pre-Assessment

What do you already know?

Return to 

Table of 

Contents

slide5

1

Evaluate the expression for a=2 and 
b=6.

b - a

slide6

2

Compare and order. Write <, > or =.

1 [ ] 2

3 5  

A

<

B

>

C

=

slide7

3

Simplify the expression by combining like 
terms.

 4x - 5 + 3x + 2

A

4x

B

12x + 2

C

7x - 3

D

7x + 7

slide8

4

Simplify the expression

(3 - d)5

A

3 - 5d

B

15 - 5d

C

15 + 5d

D

15 - d

slide9

5

Solve r - 8 = -13

slide10

6

There are 461 students and 20 teachers taking buses on a trip to a 
museum. Each bus can seat a maximum of 52. What is the least number of 
buses needed for the trip?

A

8

B

9

C

10

D

11

slide11

What do these symbols mean?

Less

Than

Greater

Than

Less Than

or Equal To

Greater Than

or Equal To

move square to reveal answer

slide12

An inequality is a statement that two quantities 
are not equal. The quantities are compared by 
using one of the following signs:

slide13

When am I ever going to use it?

Your parents and grandparents want you to start 
eating a healthy breakfast. The table shows the 
nutritional requirements for a healthy breakfast 
cereal with milk.

Did you compare your favorite 
cereal\'s nutritional values to the 
healthy requirements on the table?

Answer

Healthy Breakfast Cereals (per serving)

If you did, you found out that you 
have been eating a healthy 
breakfast. Now you can prove it 
to your parents and 
grandparents.

1. Suppose your favorite cereal has 2 grams of fat, 
7 grams of protein, 3 grams of fiber and 4 grams of 
sugar. Is it a healthy cereal?

slide14

Healthy Breakfast Cereals (per serving)

A cereal with 3 grams of fiber just 
makes it at being healthy. It needed 
at least 3 grams.

Answer

2. Is a cereal with 3 grams of fiber considered 
healthy?

slide15

Healthy Breakfast Cereals (per serving)

Answer

A cereal with 5 grams of sugar is 
still considered healthy but is 
very close to being unhealthy.

3. Is a cereal with 5 grams of sugar considered 
healthy?

slide16

Graphing and Writing

Inequalities

with One Variable

Return to 

Table of 

Contents

slide17

Graphing and Writing

Inequalities with One Variable

Objectives

Identify solutions of inequalities 
with one variable.

Write and graph inequalities with 
one variable.

slide18

When you need to use an inequality to solve a 
word problem, you may encounter one of the 
phrases below.

slide19

When you need to use an inequality to solve a word 
problem, you may encounter one of the phrases 
below.

slide20

How are these inequalities read?

2 + 2 > 3  Two plus two is greater than 3

2 + 2 > 3  Two plus two is greater than or equal to 3

2 + 2 ≥ 4  Two plus two is greater than or equal to 4

2 + 2 < 5  Two plus two is less than 5

2 + 2 ≤ 5 Two plus two is less than or equal to 5

2 + 2 ≤ 4  Two plus two is less than or equal to 4

slide21

Writing inequalities

Let\'s translate each statement into an inequality.

words

x is less than 10

translate to

inequality statement

10

x

<

20 is greater than or equal to y

20

>

y

slide22

You try a few:

1. 14 > a

2. b ≤ 8

3. 6 < 20f

4. t + 9 ≥ 36

5. 7 + w ≤ 10

6. 19 - p ≥ 2

7. n < 12

8. s < 50

9. p > 275

1. 14 is greater than a

2. b is less than or equal to 8

3. 6 is less than the product of f and 20

4. The sum of t and 9 is greater than or equal to 36

5. 7 more than w is less than or equal to 10

6. 19 decreased by p is greater than or equal to 2

7. Fewer than 12 items

8. No more than 50 students

9. At least 275 people attended the play

Answers

slide23

Do you speak math?

Try to change the following expressions from 
English into math.

Answer

2x ≤ 6

Twice a number is at most six.

Answer

2 + x ≥ 4

Two plus a number is at least four.

slide24

Three less than a number is less than three 
times that number.

x - 3 < 3x

Answer

The sum of two consecutive numbers is at 
least thirteen.

Answer

x + (x + 1) ≥ 13

Three times a number plus one is at least ten.

3x + 1 > 10

Answer

slide25

Solution Sets

A solution to an inequality is NOT a 
single number. It will have more 
than one value.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

This would be read as the solution set 
is all numbers greater than or equal to 
negative 5.

slide26

Let\'s name the numbers that are solutions of the 
given inequality.

r > 10

Which of the following are solutions? {5, 10, 15, 20}

10 > 10 is not true

So, not a solution

5 > 10 is not true

So, not a solution

15 > 10 is true

So, 15 is a solution

20 > 10 is true

So, 20 is a solution

Answer:

{15, 20} are solutions of the inequality r > 10

slide27

Let\'s try another one.

30 ≥ 5d; {4,5,6,7,8}

30 ≥ 5d

30 ≥ 5(5)

30 ≥ 25

30 ≥ 5d

30 ≥ 5(4)

30 ≥ 20

30 ≥ 5d

30 ≥ 5(6)

30 ≥ 30

30 ≥ 5d

30 ≥ 5(7)

30 ≥ 35

30 ≥ 5d

30 ≥ 5(8)

30 ≥ 40

Answer: {4,5,6}

slide28

Graphing Inequalities with 
Greater/Less Than or Equal To

An open circle on a number shows that the 
number is not part of the solution.

It is used with "greater than" and "less than".

The word equal is not included.

< >

A closed circle on a number shows that the 
number is part of the solution.

It is used with "greater than or equal to" and 
"less than or equal to".

< >

slide29

Remember!

Open circle means that 
number is not included in the 
solution set and is used to 
represent < or >.

Closed circle means the solution 
set includes that number and is 
used to represent ≤ or ≥.

slide30

Graphing Inequalities

Do you know where to start?

How do represent the starting point?

Is there a special symbol?

slide31

Graphing Inequalities

Step 1: Figure out what the inequality solution 
requires. For example, rewrite x is less than one 
as x < 1.

Step 2: Draw a circle on the number line where 
the number being graphed is represented. In 
this case, an open circle since it represents the 
starting point for the inequality solution but is 
not part of the solution.

-5

5

-4

-3

-2

-1

1

2

0

3

4

slide32

x < 1

Step 3: Draw an arrow on the number line showing 
all possible solutions. For numbers greater than the 
variable, shade to the right of the boundary point. 
For numbers less than the variable, shade to the left 
of the boundary point.

-5

5

-4

-3

-2

-1

1

2

0

3

4

Step 4: Draw a line, thicker than the horizontal line, 
from the dot to the arrow. This represents all of the 
numbers that fulfill the inequality.

-5

5

-4

-3

-2

-1

1

2

0

3

4

slide33

Graphing Inequalities

Step 1: Figure out what the inequality solution 
requires. For example, rewrite x is greater than 
or equal to one as x > 1.

Step 2: Draw a circle on the number line where 
the number being graphed is represented. In 
this case, a closed circle since it represents the 
starting point for the inequality solution and is a 
part of the solution.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide34

You try

Graph the inequality

x > 5

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

Graph the inequality

-3 > x

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide35

Try these.

Graph the inequalities.

1. x > 4 
 

-5

-4

-3

-2

-1

1

2

5

0

3

4

2. x < -5

-5

-4

-3

-2

-1

1

2

5

0

3

4

slide36

Try these.

State the inequality shown.

1.  
 

-5

-4

-3

-2

-1

1

2

5

0

3

4

2.  
 

-5

-4

-3

-2

-1

1

2

5

0

3

4

slide37

7

Would this solution set be x > 4?

True

False

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide38

Remember!

Closed circle means the solution set

includes that number and is used to

represent ≤ or ≥.

Open circle means that number is not

included in the solution set and is used

to represent < or >.

slide39

8

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

x > 3

B

x < 3

C

x < 3

D

x > 3

slide40

9

5 6 7 8 9 10 11 12 13 14 15

A

11 < x

B

11 > x

C

11 > x

D

11 < x

slide41

10

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

x > -1

B

x < -1

C

x < -1

D

x > -1

slide42

11

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

-4 < x

B

-4 > x

C

-4 < x

D

-4 > x

slide43

12

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

x > 0

B

x < 0

C

x < 0

D

x > 0

slide44

A store\'s employees earn at least $7.50 per hour. 
Define a variable and write an inequality for the 
amount the employees may earn per hour. Graph 
the solutions.

Let e represent an employee\'s wages.

An employee earns

e

at least

>

$7.50

7.5

7.5

0  1  2  3  4  5  6  7  8  9  10

slide45

Try this:

The speed limit on a road is 55 miles per hour. 
Define a variable, write an inequality and graph the 
solution.

Let s = speed

0 < s < 55

Answer

-10 0 10 20 30 40 50 60

slide46

13

The sign shown below is posted in front of a roller coaster ride at the

Wadsworth County Fairgrounds. If h represents the height of a rider 
in inches, what is a correct translation of the statement on this sign?

All riders MUST be

at least 48 inches tall.

A

h < 48

B

h > 48

C

h ≤ 48

D

h ≥ 48

slide47

Simple Inequalities 
Involving Addition

and Subtraction

Return to 

Table of 

Contents

slide48

Objectives

Solve one-step inequalities by 
using addition.

Solve one-step inequalities by 
using subtraction.

slide49

Who remembers how to solve an algebraic 
equation?

x + 3 = 13

- 3 - 3

x = 10

Use the inverse of addition

Does 10 + 3 = 13

13 = 13

Be sure to check 
your answer!

slide50

Solving one-step inequalities is much like 
solving one-step equations.

To solve an inequality, you need to isolate 
the variable using the properties of 
inequalities and inverse operations.

slide51

To find the solution, isolate the variable x.

Remember, it is isolated when it appears by itself 
on one side of the equation.

12 > x + 6

slide52

Step 1: Since 6 is added to x and 
subtraction is the inverse of addition, 
subtract 6 from both sides to undo the 
addition.

12 > x + 6

- 6- 6

6 > x

slide53

Step 2: Check the computation. Substitute the end point 
of 6 for x. The end point is not included (open circle) 
since x < 6.

12 > x + 6

12 > 6 + 6

12 > 12

0  1  2  3  4  5  6  7  8  9  10

slide54

Step 3: Check the direction of the inequality. Choose 
a number from your line (such as 4) and check that it 
fits the inequality.

6 > x

6 > 4

0  1  2  3  4  5  6  7  8  9  10

slide55

Solve and graph.

A. k + 3 > -2

k + 3 > -2

- 3

-3

k > -5

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-5 is not included in solution set; 
therefore we graph with an open circle.

slide56

Solve and graph.

B. r - 9 > 2

r - 9 > 2

+ 9

+9

r > 11

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

slide57

Solve and graph.

C. 9 > w + 4

9 > w + 4

- 4

- 4

5 > w

w < 5

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide58

14

Solve the inequality and graph the solution.

n - 2 >

1

2

1

3

5

6

2

A

-5

5

-4

-3

-2

-1

1

2

0

3

4

5

6

2

B

-5

5

-4

-3

-2

-1

1

2

0

3

4

5

6

2

C

-5

5

-4

-3

-2

-1

1

2

0

3

4

5

6

2

D

-5

5

-4

-3

-2

-1

1

2

0

3

4

slide59

15

Solve the inequality and graph the solution.

2 < s + 8

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

D

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide60

16

Solve the inequality and graph the solution.

-6 + b < -4

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

D

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide61

17

Solve the inequality and graph the solution.

-5 > b - 2

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

D

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide62

18

Solve the inequality and graph the solution.

3.5 < m + 2

1.5

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

1.5

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

1.5

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

1.5

D

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide63

Simple Inequalities 
Involving Multiplication

and Division

Return to 

Table of 

Contents

slide64

Objectives

Solve one-step inequalities by 
using multiplication.

Solve one-step inequalities by 
using division.

slide65

Multiplying or Dividing by a Positive Number

3x > -27

3x > -27

3 3

x > -9

Since x is multiplied by 3, divide 
both sides by 3 for the inverse 
operation.

Remove for Graph

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide66

Solve the inequality and graph the solution.

2

3

r < 6

Since r is multiplied by 2/3,

multiply both sides by the 
reciprocal of 2/3.

(

)

(

)

3

2

2

3

3

2

r < 6

r < 9

Remove for Graph

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide67

19

4k > 24

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

D

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide68

20

-50 > 5q

A

10 > q

B

-10 < q

-10 > q

C

D

10 < q

slide69

21

X

2

< -1

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

D

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide70

3

4

22

g > 27

A

g > 36

B

g > 108

g > 36

C

g > 108

D

slide71

23

-28 > 4d

A

d > -7

B

d > -7

C

d < -7

D

d < -7

slide72

Now let\'s see what happens when we multiply 
or divide by negative numbers.

Sometimes you must multiply or divide to 
isolate the variable.

Multiplying or dividing both sides of an 
inequality by a negative number gives a 
surprising result.

slide73

1. Write down two numbers and put the 
appropriate inequality (< or >) between them.

2. Apply each rule to your original two numbers 
from step 1 and simplify. Write the correct 
inequality(< or >) between the answers.

 A. Add 4

 B. Subtract 4

 C. Multiply by 4

 D. Multiply by -5

 E. Divide by 4

 F. Divide by -4

slide74

3. What happened with the inequality symbol in 
your results?

4. Compare your results with the rest of the 
class.

5. What pattern(s) do you notice in the 
inequalities?

How do different operations affect inequalities?

Write a rule for inequalities.

slide75

Let\'s see what happens when we multiply this 
inequality by -1.

5 > -1

-1 • 5 ? -1 •-1

-5 ? 1

-5 < 1

We know 5 is greater than -1

Multiply both sides by -1

Is -5 less than or greater 
than 1?

You know -5 is less than 1, so 
you should use <

What happened to the inequality symbol to keep 
the inequality statement true?

slide77

Helpful Hint

The direction of the inequality changes 
only if the number you are using to 
multiply or divide by is negative.

slide78

Solve and graph.

A.

-3y > 15

-3y< 15

-3 -3

y < -5

Dividing each side by -3 
changes the > to <.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide79

Solve and graph.

B.

7m < 21

7m < 21

7 7

m < 3

Divide each side by 7

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide80

Solve and graph.

C.

5m > -25

5m >-25

5 5

m > -5

Divide each side by 5.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide81

Solve and graph.

D. -8y > 24

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

E. 9f > 45

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide82

-r

2

< 5

(

)

(

)

-r

2

Multiply both sides by the 
reciprocal of -1/2.

-2

> 5

-2

Why did the inequality change?

r > -10

You multiplied by a negative.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide83

Try these.

Solve and graph each inequality.

1. -7h < 49

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

2. 3x > -15

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide84

Try these.

Solve and graph each inequality.

3. 7m < 21

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

a

-2

4. > -2

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide85

24

Solve and graph.

2y < -4

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide86

25

Solve and graph.

x

-1

< -4

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide87

26

Solve and graph.

-5y ≤ -25

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide88

27

Solve and graph.

n

-2

> 3

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide89

An inequality stays the same when you:

1. Add, subtract, multiply or divide by the same 
positive number on both sides

2. Add or subtract the same negative number on 
both sides

An inequality changes direction when you:

1. Multiply or divide by the same negative 
number on both sides

slide90

Solving Two-Step 
and Multiple-Step

Inequalities

Return to 

Table of 

Contents

slide91

Objectives

Solve inequalities that contain 
more than one operation.

Solve inequalities with variable 
terms on BOTH sides.

slide92

Now we\'ll solve some more complicated 
equations and inequalities

Ones that have two-step solutions because 
they involve two operations

Solving equations is like solving a puzzle. 
Keep working through the steps until you get 
the variable you\'re looking for alone on one side 
of the equation.

slide93

You can solve two step inequalities in the same way 
you solve equations.

3x - 10 ≤ 14

is solved in the same way as

3x - 10 = 14

You can add any positive or 
negative number to both 
sides of the inequality.

3x - 10 ≤ 14

+ 10 +10

3x<24

3 3

x < 8

You can multiply or divide 
both sides of an equality 
by any positive number.

slide94

REMEMBER! If you multiply or divide by a 
negative number, reverse the direction of 
the inequality symbol!

-3x ≤ 24

-3 -3

x ≥ -8

slide95

1. Solve this two-step equation.

  5 - 5x = 0

5 + -5x = 0

-5 -5

  -5x = -5

  -5 -5

x = 1

Step 1: Use additive inverse

Step 2: Use multiplicative  
  inverse

slide96

2. Solve this two-step inequality.

  26 < 3n + 1

  -1  - 1

25<3n

3 3

  8 < n  

Step 1: Use additive inverse

Step 2: Use multiplicative  
  inverse

1

3

slide97

Solve 4p - 9 ≥ 23

4p - 9 ≥ 23

+ 9 +9  Add 9 to both sides

4p ≥ 32  Divide both sides by 4

4 4 (sign stays the same)

p ≥ 8

Graph the solution { p | p ≥ 8 }

Move to reveal graph

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide98

Try these.

Solve and graph each inequality.

1. 6 - x > 3

2. -4c + 16 < 0

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide99

Try these.

Solve and graph each inequality.

3. -3y - 21 < 0

4. 22 < -5x + 18x - 4

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide100

Solve and graph the solution.

18 < 4(x + 2)

28

A

2.5 < x

B

2.5 > x

C

2.5 < x

D

2.5 > x

slide101

Solve and graph the solution.

16 - x > 7x

29

A

2 < x

B

2 > x

C

2 < x

D

2 > x

slide102

Solve and graph the solution.

8 < 5x + 3

30

A

1 < x

B

1 > x

C

1 < x

D

1 > x

slide103

Solve and graph the solution.

12 + 5x < 32

31

A

x > 4

B

x < 4

C

x < 4

D

x > 4

slide104

Solve and graph the solution.

36 > -3(x - 5)

32

A

-7 < x

B

-7 < x

-7 > x

C

-7 > x

D

slide105

33

Which graph represents the solution set for:

1 2 5

2 3 6

Question from ADP Algebra I

End-of-Course Practice Test

<

x

A

-2

2

-1

0

1

B

-2

2

-1

0

1

C

-2

2

-1

0

1

D

-2

2

-1

0

1

slide107

x + 5 < 17

34

Which value of x is in the solution set of

A

8

B

9

C

12

D

16

slide108

35

What is the solution of 3(2m − 1) ≤ 4m + 7?

A

m ≤ 5

B

m ≥ 5

C

m ≤ 4

D

m ≥ 4

slide109

36

In the set of positive integers, what is the solution set of the 
inequality

2x - 3 < 5?

A

{0,1,2,3}

B

{1,2,3}

C

{0,1,2,3,4}

D

{1,2,3,4}

slide110

37

The inequality

x + 3 < 2x - 6

A

x < – 5

6

B

x > – 5

6

C

x < 6

D

x > 6

slide111

38

Given: A = {18, 6, −3, −12}

Determine all elements of set A that are in the solution of 
the inequality 2x + 3 < −2x − 7.

3

A

18

B

6

C

-3

D

-12

slide112

Your town is having a fall carnival. Admission 
into the carnival is $3.00 and each game inside 
costs $0.25.

Write an inequality that represents the possible 
number of games that can be played if you have 
$10.00.

What is the maximum number of games that can 
be played?

Hint:

Ten dollars is the maximum amount of money that 
you have to spend at the carnival. What inequality 
symbol would be used?

slide113

ANSWER

.25x + 3 ≤ 10

.25x + 3 ≤ 10

- 3 -3

.25x ≤ 7

.25 .25

x ≤ 28

The maximum number of games 
that can be played is 28.

slide114

You have $65.00 in birthday money and want 
to buy some CDs and a DVD. Suppose a DVD 
cost $15.00 and a CD cost $12.00.

Write an inequality to find out how many CDs 
you can buy along with one DVD. Solve the 
inequality.

Hint 1

The cost of 1 DVD and the unknown number of 
CDs must be less or equal to $65.

Hint 2

How much does 1 CD cost? How would you 
express an unknown number of CDs?

slide115

Pull down the shade to see the answer.

15 + 12x ≤ 65

15 + 12x ≤ 65

-15 -15

12x ≤ 50

12x ≤ 50

12 12

x ≤ 4.16

Can you buy 0.16 of a CD?

You can buy 4 CDs and 1 DVD.

slide116

Matt was getting ready for school. He had less than 
$150 to buy school clothes. Matt bought 3 pairs of 
pants and spent $30 on snacks and other items.

How much could one pair of pants cost, if they were all 
the same price? Write an inequality.

What do you know?

Pull tab if you need help.

3x + 30 < 150

  - 30 - 30

3x < 120

3 3

x < 40

Matt has less than $150, he spent $30 
on snacks and bought 3 pairs of 
pants.

Are thinking about inequalities?

Would you represent the pants or the 
snacks with a variable?

Hint

Answer

slide117

Try These

1. You have $60 to spend on a concert. Tickets 
cost $18 each and parking is $8. Write an 
inequality to model the situation.

1. Let t = number of tickets

 18t + 8 < 60

2. 60 - 7w < 15

  - 7w < -45

  w > 6

Answers

2. If you borrow the $60 from your mom and pay 
her back at a rate of $7 per week, when will your 
debt be under $15?

3

7

slide118

Try This.

To earn an A in math class, you must earn a total of 
at least 180 points on three tests. On the first two 
tests, your scores were 58 and 59. What is the 
minimum score you must get on the third test in 
order to earn an A?

Define a variable, write an inequality and graph the 
solutions.

Let s = minimum score you must get

s + 58 + 59 > 180

s + 117 > 180

- 117 - 117

s > 63

You must score at least 63 to earn an A.

Answer

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide119

Thelma and Laura start a lawn-mowing business and buy a lawnmower for

$225. They plan to charge $15 to mow one lawn. What is the minimum 
number of lawns they need to mow if they wish to earn a profit of 
at least $750?

slide120

39

Roger is having a picnic for 78 guests. He plans to serve each guest at 
least one hot dog. If each package, p, contains eight hot dogs, which 
inequality could be used to determine how many packages of hot dogs 
Roger will need to buy?

A

p ≥ 78

B

8p ≥ 78

C

8 + p ≥ 78

D

78 − p ≥ 8

slide121

40

A school group needs a banner to carry in a 
parade. The narrowest street the parade is 
marching down measures 36 ft across, but 
some space is taken up by parked cars. The 
students have decided the banner should be 
18 ft long. There is 45 ft of trim available to 
sew around the border of the banner. What is 
the greatest possible width for the banner?

A

w < 27

B

w < 4.5

C

w < 18

D

w < 4.5

slide122

Solving 
Compound

Inequalities

Return to 

Table of 

Contents

slide123

Objectives

Solve inequalities that contain more 
than one operation.

Graph solution sets of compound 
inequalities.

slide124

Compound Inequalities

When two inequalities are combined into one 
statement by the words AND/OR, the result is 
called a compound inequality.

A solution of a compound inequality joined by 
and is any number that makes both 
inequalities true.

A solution of a compound inequality joined by 
or is any number that makes either inequality 
true.

slide125

Compound Inequalities

Here are some samples

x > -2 AND x < 3

-2 < x < 3

x ≥ -2 AND x ≤ 3

-2 ≤ x ≤ 3

-4

-3

-2

-1

0

1

2

3

4

-4

-3

-2

-1

0

1

2

3

4

NOTE: "and" means intersection, so you graph 
the intersection of the two inequalities

slide126

Compound Inequalities

Here are some additional samples

x < -2 OR x > 3

x ≤ -2 OR x ≥ 3

-4

-3

-2

-1

0

1

2

3

4

-4

-3

-2

-1

0

1

2

3

4

NOTE: "or" means union, so you graph the 
union of the two inequalities

slide127

41

Which inequality is represented in the graph below?

–5 –4 –3 –2 –1 0 1 2 3 4 5

A

– 4 < x < 2

B

– 4 x < 2

C

– 4 < x 2

D

– 4 x 2

slide128

42

Which inequality is represented in the accompanying graph?

–3  0  4

A

–3 ≤ x < 4

B

–3 ≤ x ≤ 4

C

–3 < x < 4

D

–3 < x ≤ 4

slide129

Solving Compound Inequalities 
that contain an AND statement

4 ≤ x+2 ≤ 8 is the same as writing

4 ≤ x+2 AND x+2 ≤ 8

You will need to solve both of these inequalities 
and graph their intersection.

slide130

Let\'s solve it!

4 ≤ x+2 ≤ 8

4 ≤ x+2 AND x+2 ≤ 8

4 ≤ x+2  AND x+2 ≤ 8

-2 -2 -2 -2

2 ≤ x AND x ≤ 6

2 < x < 6

Step 1 Rewrite as 2 
separate inequalities

Step 2 Solve each 
inequality for x

Step 3 Graph your 
solution

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide131

Let\'s try another one

-9 < x - 10 < 5

-9 < x-10 AND x-10 < 5

-9 < x-10 AND x-10 < 5

+10 +10 +10 +10

1 < x AND x < 15

1 < x < 15

What do I do next?

And then what?

1  3   5   7   9   11  13   15

slide132

43

Which result below is correct for this inequality:

-3 < x+2 < 7

A

1 < x < 5

B

-5 < x < 5

C

-3 > x > 5

slide133

Now let\'s look at the OR statements.

2 + r < 12 OR r + 5 > 19

Just like before, solve each one separately. 
However, with OR statements, graph their union.

2 + r < 12 OR r + 5 > 19

  -2 -2  - 5 -5

r < 10 OR r > 14

r < 10 or r > 14

8  10   12   14   16   18  20   22

slide134

Compound Inequalities in 
Applied Problems

Let\'s start off by translating the words of an 
applied problem into math.

The sum of 3 times a number and two lies 
between 8 and 11.

Pull

3x + 2

"The sum of 3 times a number and two" 
translates into what?

( Pull tab to see if you are correct...)

slide135

Here is another OR statement.

 7x ≥ 21 OR 2x ≤ -2

Solve each one separately, then graph their union.

7x ≥ 21  OR 2x ≤ -2

7 7    2 2

  x ≥ 3   OR x ≤ -1

  x ≥ 3 or x ≤ -1

-3  -1   1   3   5   7   9   11

slide136

Writing a Compound Inequality 
From a Graph

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

How would you write this?

x ≤ -6 OR x ≥ 0

Move to find out

slide137

Writing a Compound Inequality 
From a Graph

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

How would you write this?

-5 < x < 2

Move to find out

slide138

Try these.

Solve and graph the solution set.

1. -18 < 3x - 6 < -3

2. -5x + 2 > 27 or x - 3 > 2

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide139

Try these.

Solve and graph the solution set.

3. -2x - 6 > 4 or x + 5 > 8

4. -6 < 2x + 4 < 10

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide140

44

In order to be admitted for a certain ride at an amusement park, a child 
must be greater than or equal to 36 inches tall and less than 48 inches 
tall. Which graph represents these conditions?

A

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

B

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

C

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

D

24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54

slide141

45

Which graph shows the solution to this 
compound inequality?

r - 1 < 0 or r - 1 > 4

A

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

B

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

C

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

slide142

46

Which graph represents the solution set for 
2x - 4 ≤ 8 and x + 5 ≥ 7?

A

1 2 3 4

5 6 7

B

1 2 3 4

5 6 7

C

1 2 3 4

5 6 7

D

5 6 7

1 2 3 4

slide143

47

Solve -6 > -3x - 6 and -3x - 6 > 6

A

0 > x and x < -4

B

0 < x and x < -4

C

4 < x and x > -4

D

4 < x and x < -4

slide144

48

Solve 3x - 8 < 13 or -3x + 10 > 5

5

3

A

x < 7 or x >

5

3

5

3

B

x < or x >

5

3

C

x < 7 or x <

5

3

D

x < 7 or x >

slide145

49

The statement “x ≥ 4 and 2x - 4 < 6” 
is true when x is equal to

A

1

B

10

C

5

D

4

slide146

50

Write the inequality shown by the graph.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

A

x < -5 or x > 1

B

x < -5 and x > 1

C

1 < x and x > -6

D

x > -5 or x > 1

slide147

51

Write the inequality shown by the graph.

10

8

9

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

A

x > -7 or x < 3

B

x > -7 and x < 3

C

x > -7 or x > 3

D

x > -7 and x < 3

slide148

A cell phone plan offers free minutes for no 
more than 250 minutes per month. Define a 
variable and write an inequality for the possible 
number of free minutes. Graph the solution.

Let m = number of minutes

0 < m < 250

Why is zero a boundary?

It is not possible to use less 
than zero minutes. 
Therefore, zero is a second 
boundary.

About

Think

-100 -50  0  50  100  150  200  250 300 350 400

slide149

The sum of 3 times a number and 2 lies between 8 
and 11.

We found 3x + 2 but how will we translate "lies 
between 8 and 11"?

What inequality symbol will we use?

If 3x + 2 lies between 8 and 11, is it larger or smaller 
than 8?

Write an inequality. Pull tab to see if you are correct.

Pull

8 < 3x + 2 < 11

slide150

Solve the inequality.

8 < 3x + 2 < 11

- 2 - 2 - 2

6 < 3x < 9

3 3 3

2 < x < 3

slide151

The light rail train charges $2.00 a ticket. Children 
6 and under ride for free. Children over 6 and 
under 12 pay half fare and senior citizens (people 
over 65) get 25% off.

Write an inequality to describe x, the ages in years 
of all those who are eligible to receive reduced 
fares.

Read the problem over 
and write down which 
age groups receive a 
reduced fare.

Do people over 65? 
Who else?

Hint

slide152

Children who are 6 but less than 12 pay half. 
Children under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

Can we combine two groups?

What inequality symbols will we use?

Will this be an "and" inequality?

Could it be an "or" inequality?

We can combine all the children under 12. We 
would use x < 12.

For people that are 65 or older we would use x ≥ 
65.

x < 12 or x ≥ 65

Does someone age 25 get a reduced fare?

slide154

52

In 1999 a house sold for $145,000. The house 
sold again in 2009 for $211,000. Write a 
compound inequality that represents the 
different values that the house was worth 
between 1999 and 2008.

A

145,000 < H < 211,000

B

145,000 > H < 211,000

C

145,000 ≤ H ≤ 211,000

D

145,000 ≤ H ≥ 211,000

slide155

Special Cases of 
Compound 
Inequalities

Return to 

Table of 

Contents

slide156

Objectives

Recognize special cases of solution sets 
when solving compound inequalities.

Graph solution sets of no solution and 
Real Numbers.

slide157

Special Solutions

A solution of a compound inequality joined by 
and is any number that makes both inequalities 
true.

When there is no number that makes both 
inequalities true, we say there is no solution.

When all numbers make both inequalities true, 
we say the solution is the set of Real Numbers.

slide158

No Solution and

the Set of Real Numbers

2x > 18 AND -3x > 12 

2x >18 AND -3x > 12

2 2 -3 -3

x > 9 AND x < -4

The solution set is No Solution since there are no 
numbers that are both > 9 and < -4.

We write this solution as { } or 0

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Another Example

-2x + 3 > 17 OR 5(x + 2) > -40 

-2x + 3 > 17 OR 5x + 10 > -40

- 3 - 3 - 10 -10

-2x>14 OR 5x > -50

-2 -2  5 5

x < -7  x > -10

The solution set is Reals since all numbers are 
either < -7 or > -10.

We write this solution set as R.

-11   -10 -9 -8 -7 -6 -5 -4 -3

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Try these.

1. 4(x + 3) < 8x - 12 and 2(x + 3) < x + 6 
 

2. -2(x - 2) < 10 or 5x + 7 < 3(5 + x)

3. 3x + 8 > 23 and -2(x - 2) > -14

4. 6x + 3 > 4x - 13 and 5x + 8 > 2(x + 19)

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