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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.


Correct answer : (1)
 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.


Correct answer : (3)
 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}.


Correct answer : (3)
 4.  
Solve the inequality | 2x - 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}


Correct answer : (3)
 5.  
Solve the inequality, | 2x | < 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}


Correct answer : (2)
 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}



Correct answer : (3)
 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}


Correct answer : (1)
 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}


Correct answer : (4)
 9.  
Solve the inequality, | 2x + 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}


Correct answer : (4)
 10.  
Solve the inequality, | 5x | + 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}


Correct answer : (4)

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