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Systems of Linear Inequalities

A free Algebra I lesson from the “Systems of Linear Equations and Inequalities” unit, with a worked example and practice problems including step-by-step solutions.

A system of linear inequalities is solved by graphing each inequality and finding the overlap of the shaded regions. A point is a solution only when it satisfies EVERY inequality in the system — substitute and check all of them.

What you'll learn

Why it matters: Production planning (machine time limit AND minimum output), nutrition planning (calorie ceiling AND protein floor), and budget allocation problems all use overlapping inequality regions.

Worked example

Problem. Is (3, 1) a solution of the system y > x - 2 AND y < 2x?

  1. Check first: is 1 > 3 - 2 = 1? No (strictly greater fails).
  2. Because one inequality fails, (3, 1) is NOT a solution of the system.

Answer: No

Practice problems

1. Is (2, 3) a solution of {y < 5 AND y > 0}?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is (2, 3) a solution of {y < 5 AND y > 0}?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Check: 3 < 5 yes; 3 > 0 yes.
  4. Both hold, so (2, 3) is a solution.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

2. Is (3, 1) a solution of {y > x - 2 AND y < 2x}?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is (3, 1) a solution of {y > x - 2 AND y < 2x}?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Check first: 1 > 3 - 2 = 1 fails.
  4. Strict > is not satisfied, so not a solution.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: No

3. Is (1, 2) a solution of {y > x AND y < 2x + 1}?

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Is (1, 2) a solution of {y > x AND y < 2x + 1}?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Check: 2 > 1 yes; 2 < 3 yes.
  4. Both hold.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

4. Is (0, 0) a solution of {y >= 0 AND x >= 0}?

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: Is (0, 0) a solution of {y >= 0 AND x >= 0}?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Both 0 >= 0 statements hold.
  4. So (0, 0) is on the boundary of both and counts.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

5. Is (-1, 2) a solution of {y > 0 AND x > 0}?

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: Is (-1, 2) a solution of {y > 0 AND x > 0}?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. First: 2 > 0 yes. Second: -1 > 0 NO.
  4. Fails the x condition.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: No

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