# Absolute Value Inequalities Worksheet

Absolute Value Inequalities Worksheet
• Page 1
1.
The radius of a rod is 2.5 cm, with a tolerance of 0.003 cm. Find the least possible radius that is acceptable?
 a. 2.497 cm b. 2.5 cm c. 2.503 cm d. none of the above

#### Solution:

Let r be the radius of given part.

The tolerance limit can be expressed as an absolute value inequality: | r - 2.5 | ≤ 0.003

(r - 2.5) ≤ 0.003 or (r - 2.5) ≥ - 0.003

r ≤ 2.503 or r ≥ 2.497

So, the least possible radius that is acceptable is 2.497 cm.

2.
The width of a screw is to be 20.5 mm, with a tolerance of 0.01 mm. What is the greatest possible width that is acceptable?
 a. 20.5 mm b. 20.49 mm c. 20.51 mm d. none of the above

#### Solution:

The tolerance limit can be expressed as an absolute value inequality | n - 20.5 | ≤ 0.01

n - 20.5 ≤ 0.01 or n - 20.5 ≥ - 0.01

n ≤ 20.51 or n ≥ 20.49

So, the greatest possible width acceptable is 20.51 mm.

3.
Solve the inequality, | $x$ | + 3 < 10.
 a. { $x$: 7 < $x$ < 13 } b. { $x$: - 13 < $x$ < 13 } c. { $x$: - 7 < $x$ < 7 } d. none of the above

#### Solution:

| x | + 3 < 10

| x | < 7
[Subtracting 3 from the two sides of the equation.]

x < 7 and x > -7
[Write the equivalent conjunction.]

The solution set is {x: -7 < x < 7}.

4.
Solve the inequality | 2$x$ - 3 | - 5 > 0.
 a. { $x$: $x$ = 1 } b. { $x$: $x$ < 4 or $x$ > - 1 } c. { $x$: $x$ < -1 or $x$ > 4 } d. { $x$: $x$ > 4 }

#### Solution:

| 2x - 3 | - 5 > 0

| 2x - 3 | > 5
[Add 5 to both sides of the equation.]

2x - 3 > 5 or 2x - 3 < - 5
[Write the equivalent disjunction.]

2x > 8 or 2x < -2

x > 4 or x < -1

The solution set is: {x: x < -1 or x > 4}

5.
Solve the inequality, | 2$x$ | < 16.
 a. { $x$: - 8 ≤ $x$ ≤ 8 } b. { $x$: - 8 < $x$ < 8 } c. { $x$: - 8 < $x$ < 0 } d. { $x$: 0 < $x$ < 8 }

#### Solution:

| 2x | < 16

2x < 16 and 2x > -16

x < 8 and x > - 8
[Divide throughout by 2 and write the equivalent conjunction.]

The solution set is {x: - 8 < x < 8}

6.
Solve the inequality, | $x$ + 2 | - 3 < 5.
 a. { $x$: 6 < $x$ < 10 } b. { $x$: - 10 < $x$ < - 6 } c. { $x$: - 10 < $x$ < 6 } d. { $x$: - 2 < $x$ < 8 }

#### Solution:

| x + 2 | - 3 < 5

| x + 2 | < 8
[Add 3 to both sides of the equation.]

x + 2 < 8 and x + 2 > - 8
[Write the equivalent conjunction.]

x < 6 and x > - 10

The solution set is {x: - 10 < x < 6}

7.
Solve the inequality, | $x$ - 3 | > 4.
 a. { $x$: $x$ < -1 or $x$ > 7 } b. { $x$: $x$ < - 7 or $x$ > 1 } c. { $x$: $x$ < 1 or $x$ > 7 } d. none of the above

#### Solution:

| x - 3 | > 4

x - 3 > 4 or x - 3 < - 4
[Write the equivalent disjunction.]

x > 7 or x < - 1

The solution set is: {x: x < - 1 or x > 7}

8.
Solve the inequality, | 9 - $x$ | < 5.
 a. { $x$: $x$ < 4 or $x$ > 14 } b. { $x$: - 14 < $x$ < 4 } c. { $x$: $x$ < - 14 or $x$ > 4 } d. { $x$: 4 < $x$ < 14 }

#### Solution:

| 9 - x | < 5

9 - x < 5 and 9 - x > -5
[Write the equivalent conjunction.]

- x < - 4 and - x > - 14

x > 4 and x < 14

The solution set is: {x: 4 < x < 14}

9.
Solve the inequality, | 2$x$ + 1 | ≥ 7.
 a. { $x$: $x$ ≤ - 3 or $x$ ≥ 4 } b. { $x$: - 4 < $x$ < 3 } c. { $x$: 3 < $x$ < 4 } d. { $x$: $x$ ≤ - 4 or $x$ ≥ 3 }

#### Solution:

| 2x + 1 | ≥ 7

2x + 1 ≥ 7 or 2x + 1 ≤ -7
[Write the equivalent disjunction.]

2x ≥ 6 or 2x ≤ - 8

x ≥ 3 or x ≤ - 4

The solution set is: {x: x ≤ - 4 or x ≥ 3}

10.
Solve the inequality, | 5$x$ | + 5 ≤ 20.
 a. { $x$: $x$ ≤- 3 or $x$ ≥ 3 } b. { $x$: - 3 ≤ $x$ ≤ 0 } c. { $x$: 0 ≤ $x$ ≤ 3 } d. { $x$: - 3 ≤ $x$ ≤ 3 }

#### Solution:

| 5x | + 5 ≤ 20

| 5x | ≤ 15
[Subtracting 5 from the two sides of the equation.]

5x ≤ 15 and 5x ≥ - 15
[Write the equivalent conjunction.]

x ≤ 3 and x ≥ - 3

The solution set is: {x: - 3 ≤ x ≤ 3}