# Compound Inequalities Word problems

Compound Inequalities Word problems
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1.
Which of the following choices describes the solution of the inequality 2 < $x$ + 5 < 6?
 a. $x$ is greater than -3 and less than 1 b. $x$ is greater than 3 and less than 1 c. $x$ is greater than -3 and less than - 1 d. $x$ is greater than 3 and less than -1

#### Solution:

2 < x + 5 < 6
[Original inequality.]

2 - 5 < x + 5 - 5 < 6 - 5
[Subtract 5 from each expression.]

-3 < x < 1
[Simplify.]

The compound inequality may be described as 'x is greater than -3 and less than 1'.

2.
Which of the following best describes the solution of the inequality -3 < $x$ + 4 ≤ 5?
 a. $x$ greater than or equal to -7 and less than 1 b. $x$ greater than -7 and less than or equal to 1 c. $x$ greater than -7 and greater than or equal to 1 d. $x$ less than -7 and less than or equal to 1

#### Solution:

-3 < x + 4 ≤ 5
[Original inequality.]

-3 - 4 < x + 4 - 4 ≤ 5 - 4
[Subtract 4 from each expression.]

-7 < x ≤ 1
[Simplify.]

The compound inequality may be described as 'x greater than -7 and less than or equal to 1'.

3.
Which of the following choices describes the solution of the inequality 0 < $x$ + 7 < 9?
 a. $x$ is greater than -7 and less than 2 b. $x$ is greater than -7 and less than - 2 c. $x$ is greater than 7 and less than 2 d. $x$ is greater than 7 and less than - 2

#### Solution:

0 < x + 7 < 9
[Original inequality.]

0 - 7 < x + 7 - 7 < 9 - 7
[Subtract 7 from each expression.]

-7 < x < 2
[Simplify.]

The compound inequality may be described as 'x is greater than -7 and less than 2'.

4.
The audible frequency range of a dog is 25 Hz to 50000 Hz. Write an inequality to describe the frequency range of $f$.
 a. 25 ≤ $f$ ≤ 50000 b. 25 < $f$ < 50000 c. 25 < $f$ ≤ 50000 d. 25 ≤ $f$ < 50000

#### Solution:

The audible frequency range of a dog is from 25 Hz to 50000 Hz, that is f is greater than or equal to 25 and less than or equal to 50,000 hertz.

f ≥ 25 and f ≤ 50000
[Express in the form of inequalities.]

25 ≤ f ≤ 50000
[Combine into a single inequality.]

So, the compound inequality 25 ≤ f ≤ 50000 represents the audible frequency range of a dog.

5.
Which of the following compound inequalities represents the set of real numbers greater than or equal to - 3 and less than 3?
 a. -3 ≤ $x$ ≤ 3 b. -3 ≤ $x$ < 3 c. -3 < $x$ ≤ 3 d. -3 < $x$ < 3

#### Solution:

x ≥ -3 and x < 3
[Expressing the statement as two inequalities.]

-3 ≤ x < 3
[Combine into single inequality.]

6.
Which of the following compound inequalities represents the set of real numbers less than 6 and greater than 1?
 a. 1 ≤ $x$ < 6 b. 1 < $x$ ≤ 6 c. 1 ≤ $x$ ≤ 6 d. 1 < $x$ < 6

#### Solution:

1 < x and x < 6
[Express the statement as two inequalities.]

1 < x < 6
[Combine into single inequality.]

7.
The oxygen percent decreases as the elevation increases. If $h$ represents the elevation, find the oxygen percentage that shows the compound inequality 25000 < $h$ ≤ 50000 from the table.

 a. 14% b. 6% c. 50% d. 35%

#### Solution:

25000 < h ≤ 50000 can be read as 'the altitude greater than 25000 ft and less than or equal to 50000 ft'.

The percentage of oxygen for the above range of elevation from the table is 6%.

8.
If $y$ represents the elevation in feet, write a compound inequality that describes the approximate elevation range for 35% of oxygen.

 a. 10000 > $y$ ≥ 25000 b. 10000 < $y$ ≤ 25000 c. 10000 ≤ $y$ ≤ 25000 d. 10000 < $y$ ≥ 25000

#### Solution:

From the table 35 % of oxygen can be written in two inequalities as 1000 ≤ y and y < 25000.

1000 ≤ y < 25000
[Combine into single inequality.]

9.
Which of the follwoing choices describes the solution of the inequality 1 ≤ 3$x$ - 2 ≤ 4?
 a. $x$ is greater than or equal to 1 and less than or equal to 2 b. $x$ is greater than or equal to 1 and less than or equal to 3 c. $x$ is greater than or equal to1 and less than or equal to 4 d. None of the above

#### Solution:

1 ≤ 3x - 2 ≤ 4
[Original inequality.]

1 + 2 ≤ 3x - 2 + 2 ≤ 4 + 2

3 ≤ 3x ≤ 6
[Simplify.]

333x363
[Divide each expression by 3.]

1 ≤ x ≤ 2.
[Simplify.]

The compound inequality in verbal sentence is 'x is greater than or equal to 1 and less than or equal to 2'.

10.
Write a verbal sentence that describes the simplified inequality -3 ≤ -$x$ + 3 < 5.
 a. $x$ is greater than or equal to 2 and less than -6 b. $x$ is greater than -2 and less than or equal to 6 c. $x$ is greater than or equal to -2 and less than or equal to 6 d. None of the above

#### Solution:

-3 ≤ -x + 3 < 5
[Original inequality.]

-3 - 3 ≤ -x + 3 - 3 < 5 - 3
[Subtract 3 from each expression.]

-6 ≤ -x < 2
[Simplify.]

-1(-6) ≥ -1(-x) > -1(2)
[Multiply each expression by -1 and reverse both inequalities.]

6 ≥ x > -2.
[Simplify.]

The compound inequality in verbal sentence is 'x is greater than -2 and less than or equal to 6.