Worksheet: Absolute Value Inequalities

Practice

In this worksheet, we will practice solving inequalities that contain absolute values.

Q1:

Find the solution set of the inequality | 𝑥 + 4 | < 9 .

  • A ( − 5 , 1 3 )
  • B ( − ∞ , 5 )
  • C { − 1 3 , 5 }
  • D ( − 1 3 , 5 )
  • E ( 1 3 , ∞ )

Q2:

Find algebraically the solution set of the inequality .

  • A
  • B
  • C
  • D

Q3:

Find algebraically the solution set of the inequality | 6 − 𝑥 | < 3 .

  • A ] − 9 , − 3 [
  • B ℝ − [ 3 , 9 ]
  • C ] 3 , ∞ [
  • D ] 3 , 9 [

Q4:

Find the solution set of the inequality | 𝑥 − 6 | ≥ 7 .

  • A ℝ − [ − 1 3 , 1 ]
  • B [ 1 3 , ∞ )
  • C ℝ − { − 1 , 1 3 }
  • D ℝ − ( − 1 , 1 3 )
  • E [ 1 , ∞ )

Q5:

A body was moving with a uniform velocity of magnitude 5 cm/s from the point 𝐴 to the point 𝐶 passing through the point 𝐵 without stopping. The distance between the body and the point 𝐵 is given by 𝑑 ( 𝑡 ) = 5 | 8 − 𝑡 | , where 𝑡 is the time in seconds, and 𝑑 is the distance in cm. Determine the distance between the body and the point 𝐵 after 5 seconds and after 11 seconds.

  • A 5 cm, 5 cm
  • B 15 cm, 5 cm
  • C 3 cm, 40 cm
  • D 15 cm, 15 cm

Q6:

A body moved from position 𝐴 to position 𝐶 passing through position 𝐵 with a uniform velocity of 3 cm/s and without stopping. If the distance between the body and position 𝐵 is given by 𝑑 ( 𝑡 ) = 3 | 7 − 𝑡 | , where 𝑡 is the time in seconds, and 𝑑 is the distance in centimetres, determine the time interval during which the body is less than 9 cm from 𝐵 .

  • A ( 1 0 , ∞ )
  • B [ 4 , 1 0 ]
  • C ( 4 , ∞ )
  • D ( 4 , 1 0 )

Q7:

Find algebraically the solution set of the inequality | 8 − 𝑥 | > 1 7 .

  • A ℝ − [ − 2 5 , 9 ]
  • B ] − 9 , 2 5 [
  • C ] − ∞ , − 9 [
  • D ℝ − [ − 9 , 2 5 ]

Q8:

A factory produces cans with weight 𝑥 grams. To control the production quality, the cans are only allowed to be sold if | 𝑥 − 1 8 3 | ≤ 6 . Determine the heaviest and the lightest weight of a can that can be sold.

  • A 183 g, 177 g
  • B 189 g, 183 g
  • C 183 g, 6 g
  • D 189 g, 177 g

Q9:

Which of the following represents the interpretation for | − 3 . 3 − 𝑎 | > 5 ?

  • AThe distance between 5 and − 3 . 3 is greater than 𝑎 .
  • BThe distance between − 3 . 3 and − 𝑎 is greater than 5.
  • CThe distance between − 3 . 3 and − 𝑎 is less than 5.
  • DThe distance between − 3 . 3 and 𝑎 is greater than 5.
  • EThe distance between − 3 . 3 and 𝑎 is less than 5.

Q10:

Find the solution set of the inequality | 𝑥 − 3 | ≤ 7 .

  • A [ − 1 0 , 4 ]
  • B ( − ∞ , 1 0 ]
  • C { − 4 , 1 0 }
  • D [ − 4 , 1 0 ]
  • E [ 4 , ∞ )

Q11:

Find algebraically the solution set of the inequality .

  • A
  • B
  • C
  • D

Q12:

Find the solution set of the inequality | 𝑥 + 5 | > − 7 .

  • A ℝ − [ − 2 , 1 2 ]
  • B ( − 1 2 , ∞ )
  • C ℝ − { − 1 2 , 2 }
  • D ℝ
  • E ( − 2 , ∞ )

Q13:

Suppose that | 2 𝑥 − 𝑏 | < 2 and 𝑏 > 6 . Which of the following is true?

  • A 2 < 𝑥 < 4
  • B 𝑥 ≥ 2
  • C 2 < 𝑥 ≤ 4
  • D 𝑥 > 2
  • E 𝑥 < 4

Q14:

Suppose 𝑎 < 0 < 𝑏 . If 𝑏 > 2 , which of the following statements is true?

  • A | 𝑎 − 𝑏 | < 2
  • B | 𝑎 + 𝑏 | > 1
  • C | 𝑎 + 𝑏 | > 2
  • D | 𝑎 − 𝑏 | > 2
  • E 𝑎 − 𝑏 > − 2

Q15:

Solve | 𝑥 − 6 | ≤ 5 .

  • A 1 ≤ 𝑥 ≥ 1 1
  • B − 1 ≤ 𝑥 ≤ − 1 1
  • C 0 ≤ 𝑥 ≤ 1 0
  • D 1 ≤ 𝑥 ≤ 1 1

Q16:

Solve 𝑥 − 4 < | 2 − 𝑥 | .

  • A 𝑥 < 3
  • B 𝑥 < − 4
  • C no solution
  • Dany number
  • E − 2

Q17:

What is the interval which represents the set of all real numbers that are less than or equal to | 6 | ?

  • A ] − ∞ , 6 [
  • B [ 6 , ∞ [
  • C ] 6 , ∞ [
  • D ] − ∞ , 6 ]

Q18:

Which of the following is true?

  • A ( 1 9 + ( − 1 9 ) ) × 9 = ( | 2 7 | + ( − 2 8 ) ) × | − 9 |
  • B ( 1 9 + ( − 1 9 ) ) × 9 < ( | 2 7 | + ( − 2 8 ) ) × | − 9 |
  • C ( 1 9 + ( − 1 9 ) ) × 9 > ( | 2 7 | + ( − 2 8 ) ) × | − 9 |

Q19:

Find algebraically the solution set of the inequality | − 3 − 2 𝑥 | + | 2 𝑥 + 3 | < 2 6 .

  • A ] − 5 , 8 [
  • B ℝ − [ − 8 , 5 ]
  • C ] − ∞ , 5 [
  • D ] − 8 , 5 [

Q20:

Find algebraically the solution set of the inequality | 3 − 𝑥 | + | 2 𝑥 − 6 | ≥ 5 7 .

  • A ℝ − ] − 2 2 , 1 6 [
  • B [ − 1 6 , 2 2 ]
  • C [ 2 2 , ∞ [
  • D ℝ − ] − 1 6 , 2 2 [

Q21:

Use the graph to find the solution set of the inequality 𝑓 ( 𝑥 ) ≥ 𝑔 ( 𝑥 ) .

  • A ] − 4 , 2 [
  • B ℝ − [ − 4 , 2 ]
  • C [ − 4 , 2 ]
  • D ℝ − ] − 4 , 2 [

Q22:

Find the solution set of the inequality | 𝑥 + 1 | < 6 .

  • A ( − 5 , 7 )
  • B ( − ∞ , 5 )
  • C { − 7 , 5 }
  • D ( − 7 , 5 )
  • E ( 7 , ∞ )

Q23:

Find the solution set of the inequality | 𝑥 + 4 | < 7 .

  • A ( − 3 , 1 1 )
  • B ( − ∞ , 3 )
  • C { − 1 1 , 3 }
  • D ( − 1 1 , 3 )
  • E ( 1 1 , ∞ )

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.