Lesson Worksheet: Two-Variable Absolute Value Inequalities Mathematics

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?

  • A𝐴
  • B𝐶
  • C𝐷
  • D𝐵

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?

  • A𝑦≥|||−15𝑥(𝑥−4)|||,𝑦<|(𝑥−2)(𝑥−4)|
  • B𝑦≤|𝑥(𝑥−4)|,𝑦>|(𝑥+2)(𝑥+4)|
  • C𝑦≤|||−15𝑥(𝑥+4)|||,𝑦>|(𝑥+2)(𝑥+4)|
  • D𝑦≤|||−15𝑥(𝑥−4)|||,𝑦>|(𝑥−2)(𝑥−4)|
  • E𝑦≤|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.

  • A𝑦≥(𝑥+3)−1, 𝑦<2|𝑥+3|+1
  • B𝑦≥(𝑥−3)+1, 𝑦<2|𝑥−3|+1
  • C𝑦≥(𝑥+3)+1, 𝑦<2|𝑥−3|+1
  • D𝑦≥(𝑥−3)+1, 𝑦<2|𝑥+3|+1
  • E𝑦≥(𝑥−3)+1, 𝑦<2|𝑥+3|−1

Q7:

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

  • A𝑦>|𝑥−3|
  • B𝑦<|𝑥+3|
  • C𝑦>|𝑥+3|
  • D𝑦<|𝑥−3|
  • E𝑦≥|𝑥−3|

Q8:

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

  • A𝑦>||(𝑥+2)||
  • B𝑦≥||(𝑥−2)||
  • C𝑦<||(𝑥+2)||
  • D𝑦>||(𝑥−2)||
  • E𝑦<||(𝑥−2)||

Q9:

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

  • A𝑦≤||(𝑥−2)−4||
  • B𝑦≤||(𝑥+2)−4||
  • C𝑦≤||(𝑥−2)+4||
  • D𝑦≤||(𝑥−2)−4||
  • E𝑦≤||(𝑥+2)−4||

Q10:

Which of the following diagrams represents the shaded area represented by the inequality 𝑦≤|𝑥−4|?

  • A
  • B
  • C
  • D
  • E

Q11:

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

  • A𝑥>2
  • B𝑥<4
  • C𝑥≥2
  • D2<𝑥≤4
  • E2<𝑥<4

Q12:

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

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

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