Worksheet: Solving Systems of 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.

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0
  • B 𝑥 ≤ 0 , 𝑦 ≤ 0
  • C 𝑥 ≥ 0 , 𝑦 ≤ 0
  • D 𝑥 ≤ 0 , 𝑦 ≥ 0

Q2:

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

  • A 𝑦 ≥ 3 𝑥 − 3 , 𝑦 > − 3 𝑥 + 8
  • B 𝑦 ≥ 3 𝑥 + 3 , 𝑦 < − 3 𝑥 − 8
  • C 𝑦 ≥ 3 𝑥 + 3 , 𝑦 > − 3 𝑥 − 8
  • D 𝑦 < 3 𝑥 − 3 , 𝑦 ≤ − 3 𝑥 + 8
  • E 𝑦 > − 3 𝑥 + 3 , 𝑦 ≥ 3 𝑥 − 8

Q3:

Use the graph to find the solution set of the inequality 𝑓 ( 𝑥 ) < 𝑔 ( 𝑥 ) in ℝ .

  • A [ 0 , 6 ]
  • B ℝ − [ 0 , 6 ]
  • C ℝ − ( 0 , 6 )
  • D ( 0 , 6 )

Q4:

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

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

Q5:

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

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 ≤ 𝑥 ≤ 6 , 2 ≤ 𝑦 < 6 , 𝑥 + 𝑦 < 8
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 ≤ 𝑥 < 6 , 2 ≤ 𝑦 ≤ 6 , 𝑥 + 𝑦 > 8
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 ≤ 𝑦 < 6 , 2 ≤ 𝑥 ≤ 6 , 𝑥 + 𝑦 ≤ 8
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 ≤ 𝑥 ≤ 6 , 2 ≤ 𝑦 ≤ 6 , 𝑥 + 𝑦 ≤ 8
  • E 𝑥 ≥ 0 , 𝑦 ≥ 0 , 3 ≤ 𝑥 < 6 , 2 ≤ 𝑦 ≤ 6 , 𝑥 + 𝑦 ≤ 8

Q6:

Find the solution set of the linear inequalities shown below.

  • A ℝ
  • B 𝑦 < − 0 . 5 𝑥 − 5
  • C ∅
  • D 𝑦 > − 0 . 5 𝑥

Q7:

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

  • A C and F
  • B A and B
  • C C and D
  • D E and F
  • E D and E

Q8:

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

  • A A and D
  • B A and B
  • C D and E
  • D A and G
  • E C and D

Q9:

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

  • A F
  • B G
  • C E
  • D A
  • E D

Q10:

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

Q11:

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

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

Q12:

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

  • A A , D , E , and G
  • B A , B , D , and G
  • C A , B , D , and F
  • D C , D , E , and F
  • E A , B , C , and D

Q13:

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

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

Q14:

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

  • A E
  • B D
  • C A
  • D C
  • E F

Q15:

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

  • A A
  • B C
  • C G
  • D E
  • E B

Q16:

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

  • A A
  • B B
  • C C
  • D F
  • E E

Q17:

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

  • A 𝑦 < 2 𝑥 + 6 , 𝑦 ≥ − 𝑥 , 𝑦 ≥ − 1 0 𝑥 + 1 8
  • B 𝑦 < 2 𝑥 + 6 , 𝑦 ≥ − 𝑥 , 𝑦 ≤ 1 0 𝑥 + 1 8
  • C 𝑦 < 2 𝑥 + 6 , 𝑦 ≥ − 𝑥 , 𝑦 ≤ − 1 0 𝑥 + 1 8
  • D 𝑦 > 2 𝑥 + 6 , 𝑦 ≥ − 𝑥 , 𝑦 ≤ − 1 0 𝑥 + 1 8
  • E 𝑦 < 2 𝑥 + 6 , 𝑦 ≤ − 𝑥 , 𝑦 ≤ − 1 0 𝑥 + 1 8

Q18:

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

  • A 2 n d
  • B 1 s t
  • C 3 r d
  • D 4 t h

Q19:

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

  • A 2 n d
  • B 1 s t
  • C 4 t h
  • D 3 r d

Q20:

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

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑦 < 7 , 4 ≤ 𝑥 < 7 , 𝑥 + 𝑦 ≤ 9
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 ≤ 7 , 4 ≤ 𝑦 < 7 , 𝑥 + 𝑦 ≤ 9
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 ≤ 7 , 4 ≤ 𝑦 ≤ 7 , 𝑥 + 𝑦 < 9
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 < 7 , 4 ≤ 𝑦 < 7 , 𝑥 + 𝑦 > 9
  • E 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 < 7 , 4 ≤ 𝑦 < 7 , 𝑥 + 𝑦 ≤ 9

Q21:

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

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 < 5 , 3 ≤ 𝑦 < 6 , 𝑥 + 𝑦 > 7
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 ≤ 5 , 3 ≤ 𝑦 < 6 , 𝑥 + 𝑦 < 7
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 < 5 , 3 ≤ 𝑦 < 6 , 𝑥 + 𝑦 < 7
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 ≤ 5 , 3 ≤ 𝑦 ≤ 6 , 𝑥 + 𝑦 ≤ 7
  • E 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑦 < 5 , 3 ≤ 𝑥 < 6 , 𝑥 + 𝑦 < 7

Q22:

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

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 ≤ 5 , 4 ≤ 𝑦 ≤ 7 , 𝑥 + 𝑦 ≤ 9
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑦 < 5 , 4 ≤ 𝑥 ≤ 7 , 𝑥 + 𝑦 ≤ 9
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 < 5 , 4 ≤ 𝑦 ≤ 7 , 𝑥 + 𝑦 ≤ 9
  • D 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 ≤ 5 , 4 ≤ 𝑦 < 7 , 𝑥 + 𝑦 < 9
  • E 𝑥 ≥ 0 , 𝑦 ≥ 0 , 2 ≤ 𝑥 < 5 , 4 ≤ 𝑦 ≤ 7 , 𝑥 + 𝑦 > 9

Q23:

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

  • A 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 ≤ 𝑥 ≤ 8 , 2 ≤ 𝑦 < 5 , 𝑥 + 𝑦 ≤ 9
  • B 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 ≤ 𝑥 < 8 , 2 ≤ 𝑦 ≤ 5 , 𝑥 + 𝑦 < 9
  • C 𝑥 ≥ 0 , 𝑦 ≥ 0 , 4 ≤ 𝑥 ≤ 8 , 2 ≤ 𝑦 ≤ 5 , 𝑥 + 𝑦 < 9
  • D 𝑥 ≥