Worksheet: Two-Variable Absolute Value Inequalities

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In this worksheet, we will practice graphing absolute value inequalities and solving some real-life problems involving graphing them.

Q1:

Consider the system of inequalities 𝑦<|𝑥+4|,𝑦≥|−2𝑥+2|. Which of the shaded areas 𝐴, 𝐵, 𝐶, or 𝐷 represents the solution to this system?

  • A 𝐴
  • B 𝐶
  • C 𝐷
  • D 𝐵

Q2:

Consider the system of inequalities 𝑦>|3𝑥+5|,𝑦≤|5𝑥−12|.

Which of the following diagrams can be used to shade the area that represents the solution to this system of inequalities?

  • AA
  • BB
  • CC
  • DD

Which of the regions 1–6 of which graph represents the solution to our system of inequalities?

  • AA-4
  • BC-6
  • CD-2
  • DB-5
  • EC-2

Q3:

Which region represents the system of inequalities 𝑦<||(𝑥−1)||,𝑦>||(𝑥−1)−1||?

  • AB
  • BD
  • CA
  • DC

Q4:

Which of the following systems of inequalities is represented in the diagram?

  • A 𝑦 ≥ | | | − 1 5 𝑥 ( 𝑥 − 4 ) | | | , 𝑦 < | ( 𝑥 − 2 ) ( 𝑥 − 4 ) |
  • B 𝑦 ≤ | 𝑥 ( 𝑥 − 4 ) | , 𝑦 > | ( 𝑥 + 2 ) ( 𝑥 + 4 ) |
  • C 𝑦 ≤ | | | − 1 5 𝑥 ( 𝑥 + 4 ) | | | , 𝑦 > | ( 𝑥 + 2 ) ( 𝑥 + 4 ) |
  • D 𝑦 ≤ | | | − 1 5 𝑥 ( 𝑥 − 4 ) | | | , 𝑦 > | ( 𝑥 − 2 ) ( 𝑥 − 4 ) |
  • E 𝑦 ≤ | 𝑥 ( 𝑥 − 4 ) | , 𝑦 > | ( 𝑥 − 2 ) ( 𝑥 − 4 ) |

Q5:

Determine which region in the graph contains all the solutions to the following system of inequalities: 𝑦≥4|𝑥−2|+12,𝑦<−14(𝑥−3)+4,𝑦<1.5.

  • AF
  • BG
  • CB
  • DC
  • EE

Q6:

Find the system of inequalities that forms the shaded area shown in the graph.

  • A 𝑦 ≥ ( 𝑥 + 3 ) − 1 , 𝑦 < 2 | 𝑥 + 3 | + 1
  • B 𝑦 ≥ ( 𝑥 − 3 ) + 1 , 𝑦 < 2 | 𝑥 − 3 | + 1
  • C 𝑦 ≥ ( 𝑥 + 3 ) + 1 , 𝑦 < 2 | 𝑥 − 3 | + 1
  • D 𝑦 ≥ ( 𝑥 − 3 ) + 1 , 𝑦 < 2 | 𝑥 + 3 | + 1
  • E 𝑦 ≥ ( 𝑥 − 3 ) + 1 , 𝑦 < 2 | 𝑥 + 3 | − 1

Q7:

Which of the following inequalities represents the area shaded in the diagram?

  • A 𝑦 > | 𝑥 − 3 |
  • B 𝑦 < | 𝑥 + 3 |
  • C 𝑦 > | 𝑥 + 3 |
  • D 𝑦 < | 𝑥 − 3 |
  • E 𝑦 ≥ | 𝑥 − 3 |

Q8:

Which of the following inequalities represents the area shaded in the diagram?