Worksheet: One-Variable Absolute Value Inequalities

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

Q1:

Find the solution set of the inequality |π‘₯+4|<9.

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

Q2:

Find algebraically the solution set of the inequality |7βˆ’π‘₯|+3β‰€βˆ’6.

  • A ( βˆ’ 2 , 1 6 )
  • B βˆ…
  • C [ 1 6 , ∞ )
  • D ℝ βˆ’ [ βˆ’ 1 6 , 2 ]

Q3:

Find algebraically the solution set of the inequality |6βˆ’π‘₯|<3.

  • A ( βˆ’ 9 , βˆ’ 3 )
  • B ( 3 , ∞ )
  • C ℝ βˆ’ [ 3 , 9 ]
  • D ( 3 , 9 )

Q4:

Find the solution set of the inequality |π‘₯βˆ’6|β‰₯7.

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

Q5:

Find algebraically the solution set of the inequality |8βˆ’π‘₯|>17.

  • A ( βˆ’ 9 , 2 5 )
  • B ℝ βˆ’ [ βˆ’ 9 , 2 5 ]
  • C ℝ βˆ’ [ βˆ’ 2 5 , 9 ]
  • D ( βˆ’ ∞ , βˆ’ 9 )

Q6:

A factory produces cans with weight π‘₯ grams. To control the production quality, the cans are only allowed to be sold if |π‘₯βˆ’183|≀6. Determine the heaviest and the lightest weight of a can that can be sold.

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

Q7:

Which of the following represents the interpretation for |βˆ’3.3βˆ’π‘Ž|>5?

  • AThe distance between 5 and βˆ’3.3 is greater than π‘Ž.
  • BThe distance between βˆ’3.3 and π‘Ž is greater than 5.
  • CThe distance between βˆ’3.3 and βˆ’π‘Ž is greater than 5.
  • DThe distance between βˆ’3.3 and π‘Ž is less than 5.
  • EThe distance between βˆ’3.3 and βˆ’π‘Ž is less than 5.

Q8:

Find the solution set of the inequality |π‘₯βˆ’3|≀7.

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

Q9:

Find algebraically the solution set of the inequality |π‘₯βˆ’3|+|π‘₯βˆ’5|>6.

  • A ℝ βˆ’ [ 1 , 7 ]
  • B ( 1 , 7 )
  • C [ 7 , ∞ )
  • D ( βˆ’ ∞ , 1 )

Q10:

Find the solution set of the inequality |π‘₯+5|>βˆ’7.

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

Q11:

Suppose that |2π‘₯βˆ’π‘|<2 and 𝑏>6. Which of the following is true?

  • A π‘₯ > 2
  • B π‘₯ < 4
  • C π‘₯ β‰₯ 2
  • D 2 < π‘₯ ≀ 4
  • E 2 < π‘₯ < 4

Q12:

Suppose π‘Ž<0<𝑏. If 𝑏>2, which of the following statements is true?

  • A | π‘Ž + 𝑏 | > 1
  • B π‘Ž βˆ’ 𝑏 > βˆ’ 2
  • C | π‘Ž βˆ’ 𝑏 | < 2
  • D | π‘Ž βˆ’ 𝑏 | > 2
  • E | π‘Ž + 𝑏 | > 2

Q13:

Solve |π‘₯βˆ’6|≀5.

  • A 1 ≀ π‘₯ β‰₯ 1 1
  • B 1 ≀ π‘₯ ≀ 1 1
  • C 0 ≀ π‘₯ ≀ 1 0
  • D βˆ’ 1 ≀ π‘₯ ≀ βˆ’ 1 1

Q14:

Solve π‘₯βˆ’4<|2βˆ’π‘₯|.

  • A π‘₯ < 3
  • Bno solution
  • C π‘₯ < βˆ’ 4
  • D βˆ’ 2
  • Eany number

Q15:

Find algebraically the solution set of the inequality |8βˆ’π‘₯|+|π‘₯βˆ’8|≀6.

  • A ( βˆ’ ∞ , 1 1 ]
  • B ℝ βˆ’ ( 5 , 1 1 )
  • C [ 5 , 1 1 ]
  • D [ βˆ’ 1 1 , βˆ’ 5 ]

Q16:

Find algebraically the solution set of the inequality |3βˆ’π‘₯|+|2π‘₯βˆ’6|β‰₯57.

  • A ℝ βˆ’ ( βˆ’ 2 2 , 1 6 )
  • B [ βˆ’ 1 6 , 2 2 ]
  • C ℝ βˆ’ ( βˆ’ 1 6 , 2 2 )
  • D [ 2 2 , ∞ )

Q17:

Find the solution set of the inequality |π‘₯+1|<6.

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

Q18:

Find the solution set of the inequality |π‘₯+4|<7.

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

Q19:

Use the absolute value to write an inequality statement which describes all values π‘₯ that are a distance of 8 from the number 3.

  • A | π‘₯ βˆ’ 3 | ≀ 8
  • B | π‘₯ βˆ’ 8 | ≀ 3
  • C | π‘₯ βˆ’ 3 | β‰₯ 8
  • D | π‘₯ + 3 | ≀ 8

Q20:

Express βˆ’4<π‘₯<4 in the form of an absolute value inequality.

  • A | π‘₯ | > βˆ’ 4
  • B | π‘₯ | < βˆ’ 4
  • C | π‘₯ | < 4
  • D | π‘₯ | > 4

Q21:

Express the following as an absolute value inequality: π‘₯>4, π‘₯<βˆ’4.

  • A | π‘₯ | < 4
  • B | π‘₯ | > 4
  • C | π‘₯ | < βˆ’ 4
  • D | π‘₯ | > βˆ’ 4

Q22:

Given that students’ marks in an exam range from 69 to 93, write an absolute value inequality to express the range of marks.

  • A | π‘₯ | < 1 2
  • B | π‘₯ βˆ’ 8 1 | < 1 2
  • C | π‘₯ βˆ’ 6 9 | < 9 3
  • D | π‘₯ βˆ’ 8 1 | < 8 1

Q23:

An algae in the ocean can reach a maximum depth of 138 meters. Let π‘₯ represent the depth of an algae. Express, in the form of an absolute value, the range of values that π‘₯ can take.

  • A | π‘₯ βˆ’ 6 9 | β‰₯ 6 9
  • B | π‘₯ | ≀ 1 3 8
  • C | π‘₯ | ≀ 6 9
  • D | π‘₯ βˆ’ 6 9 | ≀ 6 9

Q24:

Express 0<π‘₯<66 in the form of an absolute value inequality.

  • A | π‘₯ | < 3 3
  • B | π‘₯ | < 6 6
  • C | π‘₯ βˆ’ 6 6 | < 6 6
  • D | π‘₯ βˆ’ 3 3 | < 3 3

Q25:

Determine the absolute value inequality representing π‘₯βˆˆβ„βˆ’[βˆ’21,27].

  • A | π‘₯ βˆ’ 3 | > 2 4
  • B | π‘₯ | > 3
  • C | π‘₯ | > 2 4
  • D | π‘₯ βˆ’ 2 1 | > 2 4

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