Worksheet: Solving Systems of Linear Inequalities

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In this worksheet, we will practice solving systems of linear inequalities by graphing them and identifying the regions representing the solution.

Q1:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≤0, 𝑦≥0
  • B𝑥≥0, 𝑦≤0
  • C𝑥≤0, 𝑦≤0
  • D𝑥≥0, 𝑦≥0

Q2:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑦>−3𝑥+3, 𝑦≥3𝑥−8
  • B𝑦≥3𝑥−3, 𝑦>−3𝑥+8
  • C𝑦≥3𝑥+3, 𝑦<−3𝑥−8
  • D𝑦<3𝑥−3, 𝑦≤−3𝑥+8
  • E𝑦≥3𝑥+3, 𝑦>−3𝑥−8

Q3:

Which of the points below belongs to the solution set of the system of inequalities represented by the figure shown?

  • A(6,8)
  • B(1,3)
  • C(3,6)
  • D(4,3)

Q4:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≥0, 𝑦≥0, 3≤𝑥≤6, 2≤𝑦<6, 𝑥+𝑦<8
  • B𝑥≥0, 𝑦≥0, 3≤𝑥<6, 2≤𝑦≤6, 𝑥+𝑦>8
  • C𝑥≥0, 𝑦≥0, 3≤𝑦<6, 2≤𝑥≤6, 𝑥+𝑦≤8
  • D𝑥≥0, 𝑦≥0, 3≤𝑥<6, 2≤𝑦≤6, 𝑥+𝑦≤8
  • E𝑥≥0, 𝑦≥0, 3≤𝑥≤6, 2≤𝑦≤6, 𝑥+𝑦≤8

Q5:

Find the solution set of the linear inequalities shown below.

  • A𝑦<−0.5𝑥−5
  • B∅
  • Cℝ
  • D𝑦>−0.5𝑥

Q6:

Which regions on the graph contain solutions that satisfy both of the inequalities 𝑦<𝑥,𝑦≥2𝑥−4?

  • AA and B
  • BD and E
  • CC and F
  • DC and D
  • EE and F

Q7:

Which regions on the graph contain solutions to both of the inequalities 𝑦>−𝑥−4,𝑦≥2𝑥−4?

  • AC and D
  • BA and G
  • CA and D
  • DD and E
  • EA and B

Q8:

Which region on the graph contains solutions to the set of inequalities 𝑦>𝑥,𝑦≤2𝑥−4?

  • AE
  • BG
  • CF
  • DA
  • ED

Q9:

Use the graph to identify which of the following points is a solution to the system of inequalities 𝑦<2,𝑥≥−2,𝑦>𝑥−3.

  • A(0,−4)
  • B(−2,2)
  • C(1,−1)
  • D(3,3)
  • E(−3,2)

Q10:

Use the graph to identify which of the following points is a solution to the set of inequalities. 𝑦<2𝑥≤−2𝑦>𝑥−3

  • A(−2,−2)
  • B(3,3)
  • C(1,−1)
  • D(−3,2)
  • E(0,−4)

Q11:

Which of the regions on the graph contain solutions to the inequality 𝑦≥2𝑥−4?

  • AA, B, C, and D
  • BA, B, D, and G
  • CA, D, E, and G
  • DA, B, D, and F
  • EC, D, E, and F

Q12:

Use the graph to identify which of the following points is NOT a solution to the given set of inequalities. 𝑦<2𝑥+4𝑦≤−3𝑥𝑥≤2

  • A(0,0)
  • B(1,−3)
  • C(2,−6)
  • D(1,2)
  • E(0,−10)

Q13:

Which region on the graph contains solutions to the set of inequalities 𝑦>2,𝑦≥−𝑥,𝑥<1?

  • AA
  • BF
  • CD
  • DE
  • EC

Q14:

Which region on the graph contains solutions to the set of inequalities 𝑦<2,𝑦≤−𝑥,𝑥<1.

  • AB
  • BE
  • CG
  • DC
  • EA

Q15:

Which region on the graph contains solutions to the set of inequalities 𝑦<2,𝑦≥−𝑥,𝑥<1?

  • AB
  • BE
  • CC
  • DF
  • EA

Q16:

Find the system of inequalities that forms the triangle shown in the graph.

  • A𝑦<2𝑥+6,𝑦≤−𝑥,𝑦≤−10𝑥+18
  • B𝑦<2𝑥+6,𝑦≥−𝑥,𝑦≥−10𝑥+18
  • C𝑦<2𝑥+6,𝑦≥−𝑥,𝑦≤−10𝑥+18
  • D𝑦<2𝑥+6,𝑦≥−𝑥,𝑦≤10𝑥+18
  • E𝑦>2𝑥+6,𝑦≥−𝑥,𝑦≤−10𝑥+18

Q17:

Fill in the blank: The quadrant representing the solution set of the inequalities 𝑦<0 and 𝑥>0 is the quadrant.

  • A2nd
  • B4th
  • C1st
  • D3rd

Q18:

Fill in the blank: The quadrant representing the solution set of the inequalities 𝑦<0 and 𝑥<0 is the quadrant.

  • A3rd
  • B1st
  • C4th
  • D2nd

Q19:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≥0, 𝑦≥0, 2≤𝑥≤7, 4≤𝑦≤7, 𝑥+𝑦<9
  • B𝑥≥0, 𝑦≥0, 2≤𝑥<7, 4≤𝑦<7, 𝑥+𝑦>9
  • C𝑥≥0, 𝑦≥0, 2≤𝑦<7, 4≤𝑥<7, 𝑥+𝑦≤9
  • D𝑥≥0, 𝑦≥0, 2≤𝑥<7, 4≤𝑦<7, 𝑥+𝑦≤9
  • E𝑥≥0, 𝑦≥0, 2≤𝑥≤7, 4≤𝑦<7, 𝑥+𝑦≤9

Q20:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≥0, 𝑦≥0, 2≤𝑥≤5, 3≤𝑦≤6, 𝑥+𝑦≤7
  • B𝑥≥0, 𝑦≥0, 2≤𝑥<5, 3≤𝑦<6, 𝑥+𝑦>7
  • C𝑥≥0, 𝑦≥0, 2≤𝑦<5, 3≤𝑥<6, 𝑥+𝑦<7
  • D𝑥≥0, 𝑦≥0, 2≤𝑥<5, 3≤𝑦<6, 𝑥+𝑦<7
  • E𝑥≥0, 𝑦≥0, 2≤𝑥≤5, 3≤𝑦<6, 𝑥+𝑦<7

Q21:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≥0, 𝑦≥0, 2≤𝑥≤5, 4≤𝑦<7, 𝑥+𝑦<9
  • B𝑥≥0, 𝑦≥0, 2≤𝑥<5, 4≤𝑦≤7, 𝑥+𝑦>9
  • C𝑥≥0, 𝑦≥0, 2≤𝑦<5, 4≤𝑥≤7, 𝑥+𝑦≤9
  • D𝑥≥0, 𝑦≥0, 2≤𝑥<5, 4≤𝑦≤7, 𝑥+𝑦≤9
  • E𝑥≥0, 𝑦≥0, 2≤𝑥≤5, 4≤𝑦≤7, 𝑥+𝑦≤9

Q22:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≥0, 𝑦≥0, 4≤𝑥≤8, 2≤𝑦<5, 𝑥+𝑦≤9
  • B𝑥≥0, 𝑦≥0, 4≤𝑥<8, 2≤𝑦≤5, 𝑥+𝑦>9
  • C𝑥≥0, 𝑦≥0, 4≤𝑦<8, 2≤𝑥≤5, 𝑥+𝑦<9
  • D𝑥≥0, 𝑦≥0, 4≤𝑥<8, 2≤𝑦≤5, 𝑥+𝑦<9
  • E𝑥≥0, 𝑦≥0, 4≤𝑥≤8, 2≤𝑦≤5, 𝑥+𝑦<9

Q23:

State the system of inequalities whose solution is represented by the following graph.

  • A𝑥≥0, 𝑦≥0
  • B𝑥≥0, 𝑦≤0
  • C𝑥≤0, 𝑦≤0
  • D𝑥≤0, 𝑦≥0

Q24:

Which region on the graph contains solutions to the set of inequalities 𝑦<𝑥,𝑦≥2𝑥−4,𝑦>−𝑥−4?

  • AF
  • BE
  • CD
  • DC
  • EA

Q25:

Which region on the graph contains solutions to the set of inequalities 𝑦<𝑥,𝑦≤2𝑥−4,𝑦>−𝑥−4?

  • AA
  • BE
  • CF
  • DC
  • ED

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