﻿ Compound Inequalities Worksheet | Problems & Solutions

# Compound Inequalities Worksheet

Compound Inequalities Worksheet
• Page 1
1.
Identify the graph that represents $x$ > 5 or $x$ ≤ - 2

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

#### Solution:

x > 5 or x ≤ - 2
[Original inequality.]

x > 5 or x ≤ - 2
[Original inequality.]

So, x has the set of all real numbers that are greater than 5 or the set of all real numbers that are less than or equal to - 2.

So, x has the set of all real numbers that are greater than 5 or the set of all real numbers that are less than or equal to - 2.

Therefore, Graph 3 represents x > 5 or x ≤ - 2.
[Closed circle on the number line indicates that - 2 is included. Open circle on the number line indicates that 5 is excluded.]

Therefore, Graph 3 represents x > 5 or x ≤ - 2.
[Closed circle on the number line indicates that - 2 is included. Open circle on the number line indicates that 5 is excluded.]

2.
Which of the graphs represents the compound inequality $x$ ≤ 0 or $x$ ≥ 5?

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

#### Solution:

x ≤ 0 or x ≥ 5 is read as 'x is less than or equal to 0 or x is greater than or equal to 5'.

From Graph 1, 'x is less than or equal to 0 and greater than or equal to 5'.
[Closed circle indicates that 0 and 5 are also included.]

From Graph 2, 'x is less than 0 and greater than 5'.
[Open circle indicates that 0 and 5 are not included.]

From Graph 3, 'x is less than or equal to 0 and greater than 5'.
[Closed circle indicates that 0 is included and open circle indicates that 5 is not included.]

From Graph 4, 'x is greater than or equal to 0 and less than or equal to 5'.
[Closed circle indicates that 0 and 5 are also included.]

Comparing the inequality and graphs, Graph 1 matches the inequality.

3.
Which of the graphs best suits for 0 ≤ $x$ - 2 ≤ 7?

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

#### Solution:

0 ≤ x - 2 ≤ 7
[Original inequality.]

0 ≤ x - 2 and x - 2 ≤ 7
[Write the inequality as two inequalities.]

0 + 2 ≤ x - 2 + 2 and x - 2 + 2 ≤ 7 + 2

2 ≤ x and x ≤ 9
[Simplify.]

2 ≤ x ≤ 9
[Write compound inequality.]

The solution is all real numbers greater than or equal to 2 and less than or equal to 9. The graph of the solution can be represented as shown.

4.
Which of the graphs represents the solution of the inequality 3$x$ + 1 < 10 (or) 3$x$ - 5 ≥ 10?

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

#### Solution:

3x + 1 < 10 or 3x - 5 ≥ 10
[Original inequalities.]

3x + 1 - 1 < 10 - 1 or 3x - 5 + 5 ≥ 10 + 5
[Subtract 1 from both sides for the first inequality and add 5 on both sides for the second inequality.]

3x < 9 or 3x ≥ 15

x < 3 or x ≥ 5
[Divide both inequalities by 3 on both sides.]

The solutions are x is less than 3 or greater than or equal to 5.

Draw the solutions on the number line.

So, Graph 2 represents the solution of the inequality 3x + 1 < 10 or 3x - 5 ≥ 10.

5.
Which of the graphs represents the solution of - 8 < - 4$x$ ≤ 12?

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

#### Solution:

- 8 < - 4x ≤ 12
[Original inequality.]

- 84< - 4x4124
[Divide each expression by 4.]

- 2 < - x ≤ 3
[Simplify.]

- 1(- 2) > - 1(- x) ≥ - 1(3)
[Multiply each expression by - 1 and reverse both inequalities.]

2 > x ≥ - 3
[Simplify.]

The solution is all real numbers greater than or equal to - 3 and less than 2. The graph for the solution is:

6.
Which of the following best describes the solution of the inequality - 1 < $x$ + 2 ≤ 4?
 a. $x$ is greater than - 3 and greater than or equal to 2. b. $x$ is greater than - 3 and less than or equal to 2. c. $x$ is greater than or equal to - 3 and less than 2. d. $x$ is less than - 3 and less than or equal to 2.

#### Solution:

- 1 < x + 2 ≤ 4
[Original inequality.]

- 1 - 2 < x + 2 - 2 ≤ 4 - 2
[Subtract 2 from each expression.]

- 3 < x ≤ 2
[Simplify.]

The compound inequality may be described as 'x is greater than - 3 and less than or equal to 2'.

7.
Which of the following best describes the solution of the inequality 1 < $x$ + 3 < 6?
 a. $x$ is greater than 2 and less than 3. b. $x$ is greater than 2 and less than - 3. c. $x$ is greater than - 2 and less than 3. d. $x$ is greater than - 2 and less than - 3.

#### Solution:

1 < x + 3 < 6
[Original inequality.]

1 - 3 < x + 3 - 3 < 6 - 3
[Subtract 3 from each expression.]

- 2 < x < 3
[Simplify.]

The compound inequality may be described as 'x is greater than - 2 and less than 3'.

8.
The audible frequency range of a dog is 25 Hz to 50,000 Hz. Which of the following best describes the frequency range of $f$?
 a. 25 ≤ $f$ < 50,000 b. 25 < $f$ < 50,000 c. 25 ≤ $f$ ≤ 50,000 d. 25 < $f$ ≤ 50,000

#### Solution:

The audible frequency range of a dog is from 25 Hz to 50,000 Hz, that is f is greater than or equal to 25 and less than or equal to 50,000 hertz.

f ≥ 25 and f ≤ 50,000
[Express in the form of inequalities.]

25 ≤ f ≤ 50,000
[Write compound inequality.]

So, the compound inequality 25 ≤ f ≤ 50,000 represents the audible frequency range of a dog.

9.
Which of the following represents the set of all real numbers greater than or equal to - 4 and less than 2?
 a. - 4 < $x$ ≤ 2 b. - 4 ≤ $x$ < 2 c. - 4 < $x$ < 2 d. - 4 ≤ $x$ ≤ 2

#### Solution:

x ≥ - 4 and x < 2
[Expressing the statement as two inequalities.]

- 4 ≤ x < 2
[Write compound inequality.]

10.
Which of the following best represents the set of all real numbers less than 9 and greater than 1?
 a. 1 ≤ $x$ ≤ 9 b. 1 < $x$ < 9 c. 1 ≤ $x$ < 9 d. 1 < $x$ ≤ 9

#### Solution:

1 < x and x < 9
[Express the statement as two inequalities.]

1 < x < 9
[Write compound inequality.]