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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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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

Return to 

Table of 

Contents


Key Terms

Equation

Inequality

Solution Set

Compound Inequality

Set Builder Notation


Pre-Assessment

What do you already know?

Return to 

Table of 

Contents


1

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

b - a


2

Compare and order. Write <, > or =.

1 [ ] 2

3 5  

A

<

B

>

C

=


3

Simplify the expression by combining like 
terms.

 4x - 5 + 3x + 2

A

4x

B

12x + 2

C

7x - 3

D

7x + 7


4

Simplify the expression

(3 - d)5

A

3 - 5d

B

15 - 5d

C

15 + 5d

D

15 - d


5

Solve r - 8 = -13


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


What do these symbols mean?

Less

Than

Greater

Than

Less Than

or Equal To

Greater Than

or Equal To

move square to reveal answer


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


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?


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?


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?


Graphing and Writing

Inequalities

with One Variable

Return to 

Table of 

Contents


Graphing and Writing

Inequalities with One Variable

Objectives

Identify solutions of inequalities 
with one variable.

Write and graph inequalities with 
one variable.


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


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


How are these inequalities read? 
problem, you may encounter one of the phrases 
below.

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


Writing inequalities 
problem, you may encounter one of the phrases 
below.

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


You try a few: 
problem, you may encounter one of the phrases 
below.

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


Do you speak math? 
problem, you may encounter one of the phrases 
below.

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.


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


Solution Sets number.

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.


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


Let's try another one. inequality.

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}


Graphing Inequalities with 
Greater/Less Than or Equal To inequality.

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".

< >


Remember! inequality.

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 ≥.


Graphing Inequalities inequality.

Do you know where to start?

How do represent the starting point?

Is there a special symbol?


Graphing Inequalities inequality.

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


x < 1 inequality.

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


Graphing Inequalities inequality.

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


You try inequality.

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


Try these. inequality.

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


Try these. inequality.

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


7 inequality.

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


Remember! inequality.

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 >.


8 inequality.

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

x > 3

B

x < 3

C

x < 3

D

x > 3


9 inequality.

5 6 7 8 9 10 11 12 13 14 15

A

11 < x

B

11 > x

C

11 > x

D

11 < x


10 inequality.

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

x > -1

B

x < -1

C

x < -1

D

x > -1


11 inequality.

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

-4 < x

B

-4 > x

C

-4 < x

D

-4 > x


12 inequality.

-5

5

-4

-3

-2

-1

0

1

2

3

4

A

x > 0

B

x < 0

C

x < 0

D

x > 0


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


Try this: a variable and write an inequality for the 
amount the employees may earn per hour. Graph 
the solutions.

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


13 a variable and write an inequality for the 
amount the employees may earn per hour. Graph 
the solutions.

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


Simple Inequalities 
Involving Addition a variable and write an inequality for the 
amount the employees may earn per hour. Graph 
the solutions.

and Subtraction

Return to 

Table of 

Contents


Objectives a variable and write an inequality for the 
amount the employees may earn per hour. Graph 
the solutions.

Solve one-step inequalities by 
using addition.

Solve one-step inequalities by 
using subtraction.


Who remembers how to solve an algebraic 
equation? a variable and write an inequality for the 
amount the employees may earn per hour. Graph 
the solutions.

x + 3 = 13

- 3 - 3

x = 10

Use the inverse of addition

Does 10 + 3 = 13

13 = 13

Be sure to check 
your answer!


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.


To find the solution, isolate the variable x. one-step equations.

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

12 > x + 6


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


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


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


Solve and graph. number from your line (such as 4) and check that it 
fits the inequality.

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.


Solve and graph. number from your line (such as 4) and check that it 
fits the inequality.

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


Solve and graph. number from your line (such as 4) and check that it 
fits the inequality.

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


14 number from your line (such as 4) and check that it 
fits the inequality.

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


15 number from your line (such as 4) and check that it 
fits the inequality.

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


16 number from your line (such as 4) and check that it 
fits the inequality.

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


17 number from your line (such as 4) and check that it 
fits the inequality.

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


18 number from your line (such as 4) and check that it 
fits the inequality.

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


Simple Inequalities 
Involving Multiplication number from your line (such as 4) and check that it 
fits the inequality.

and Division

Return to 

Table of 

Contents


Objectives number from your line (such as 4) and check that it 
fits the inequality.

Solve one-step inequalities by 
using multiplication.

Solve one-step inequalities by 
using division.


Multiplying or Dividing by a Positive Number number from your line (such as 4) and check that it 
fits the inequality.

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


Solve the inequality and graph the solution. number from your line (such as 4) and check that it 
fits the inequality.

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


19 number from your line (such as 4) and check that it 
fits the inequality.

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


20 number from your line (such as 4) and check that it 
fits the inequality.

-50 > 5q

A

10 > q

B

-10 < q

-10 > q

C

D

10 < q


21 number from your line (such as 4) and check that it 
fits the inequality.

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


3 number from your line (such as 4) and check that it 
fits the inequality.

4

22

g > 27

A

g > 36

B

g > 108

g > 36

C

g > 108

D


23 number from your line (such as 4) and check that it 
fits the inequality.

-28 > 4d

A

d > -7

B

d > -7

C

d < -7

D

d < -7


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.


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


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.


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?


Helpful Hint by -1.

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


Solve and graph. by -1.

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


Solve and graph. by -1.

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


Solve and graph. by -1.

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


Solve and graph. by -1.

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


-r by -1.

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


Try these. by -1.

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


Try these. by -1.

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


24 by -1.

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


25 by -1.

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


26 by -1.

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


27 by -1.

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


An inequality by -1.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


Solving Two-Step 
and Multiple-Step by -1.

Inequalities

Return to 

Table of 

Contents


Objectives by -1.

Solve inequalities that contain 
more than one operation.

Solve inequalities with variable 
terms on BOTH sides.


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.


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.


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

-3x ≤ 24

-3 -3

x ≥ -8


1. Solve this two-step equation. number,

  5 - 5x = 0

5 + -5x = 0

-5 -5

  -5x = -5

  -5 -5

x = 1

Step 1: Use additive inverse

Step 2: Use multiplicative  
  inverse


2. Solve this two-step inequality. number,

  26 < 3n + 1

  -1  - 1

25<3n

3 3

  8 < n  

Step 1: Use additive inverse

Step 2: Use multiplicative  
  inverse

1

3


Solve 4p - 9 ≥ 23 number,

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


Try these. number,

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


Try these. number,

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


Solve and graph the solution. number,

18 < 4(x + 2)

28

A

2.5 < x

B

2.5 > x

C

2.5 < x

D

2.5 > x


Solve and graph the solution. number,

16 - x > 7x

29

A

2 < x

B

2 > x

C

2 < x

D

2 > x


Solve and graph the solution. number,

8 < 5x + 3

30

A

1 < x

B

1 > x

C

1 < x

D

1 > x


Solve and graph the solution. number,

12 + 5x < 32

31

A

x > 4

B

x < 4

C

x < 4

D

x > 4


Solve and graph the solution. number,

36 > -3(x - 5)

32

A

-7 < x

B

-7 < x

-7 > x

C

-7 > x

D


33 number,

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



x + 5 < 17 inequality:

34

Which value of x is in the solution set of

A

8

B

9

C

12

D

16


35 inequality:

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

A

m ≤ 5

B

m ≥ 5

C

m ≤ 4

D

m ≥ 4


36 inequality:

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}


37 inequality:

The inequality

x + 3 < 2x - 6

A

x < – 5

6

B

x > – 5

6

C

x < 6

D

x > 6


38 inequality:

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


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?


ANSWER carnival is $3.00 and each game inside 
costs $0.25.

.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.


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?


Pull down the shade to see the answer. CDs and a DVD. Suppose a DVD 
cost $15.00 and a CD cost $12.00.

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.


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


Try These to buy school clothes. Matt bought 3 pairs of 
pants and spent $30 on snacks and other items.

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


Try This. to buy school clothes. Matt bought 3 pairs of 
pants and spent $30 on snacks and other items.

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


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?


39 lawnmower for

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


40 lawnmower for

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


Solving 
Compound lawnmower for

Inequalities

Return to 

Table of 

Contents


Objectives lawnmower for

Solve inequalities that contain more 
than one operation.

Graph solution sets of compound 
inequalities.


Compound Inequalities lawnmower for

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.


Compound Inequalities lawnmower for

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


Compound Inequalities lawnmower for

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


41 lawnmower for

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


42 lawnmower for

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


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.


Let's solve it! statement

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


Let's try another one statement

-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


43 statement

Which result below is correct for this inequality:

-3 < x+2 < 7

A

1 < x < 5

B

-5 < x < 5

C

-3 > x > 5


Now let's look at the OR statements. statement

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


Compound Inequalities in 
Applied Problems statement

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...)


Here is another OR statement. 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


Writing a Compound Inequality 
From a Graph statement

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


Writing a Compound Inequality 
From a Graph statement

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


Try these. statement

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


Try these. statement

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


44 statement

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


45 statement

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


46 statement

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


47 statement

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


48 statement

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 >


49 statement

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

A

1

B

10

C

5

D

4


50 statement

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


51 statement

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


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


The sum of 3 times a number and 2 lies between 8 
and 11. 250 minutes per month. Define a 
variable and write an inequality for the possible 
number of free minutes. Graph the solution.

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


Solve the inequality. 250 minutes per month. Define a 
variable and write an inequality for the possible 
number of free minutes. Graph the solution.

8 < 3x + 2 < 11

- 2 - 2 - 2

6 < 3x < 9

3 3 3

2 < x < 3


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


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?


Draw a graph to illustrate the inequality. under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

x < 12 or x ≥ 65


52 under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

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


Special Cases of 
Compound 
Inequalities under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

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Objectives under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

Recognize special cases of solution sets 
when solving compound inequalities.

Graph solution sets of no solution and 
Real Numbers.


Special Solutions under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

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.


No Solution and under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

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


Another Example under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

-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


Try these. under 6 ride free. People 65 or older pay 
a reduced fare. How does that translate into an 
inequality?

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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