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Linear Inequalities

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

Solving linear inequalities is similar to solving equations, but the result is a range of values. If you multiply or divide both sides by a negative number, reverse the inequality symbol.

What you'll learn

Why it matters: Inequalities model limits such as budget caps, minimum scores, speed ranges, inventory thresholds, and safety tolerances.

Worked example

Problem. Solve -2x + 5 < 13.

  1. Subtract 5 from both sides: -2x < 8.
  2. Divide by -2 and flip the inequality.
  3. x > -4.

Answer: x > -4

Practice problems

1. Solve x + 6 > 14. Enter the boundary number.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x + 6 > 14. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 6 from both sides.
  4. x > 8.
  5. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

2. Solve 3x <= 21. Enter the boundary number.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 3x <= 21. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Divide by 3.
  4. x <= 7.
  5. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

3. Which phrase matches x >= -2?

Choices: x is at least -2 · x is below -2 · x is less than -2 · x equals -2 only

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which phrase matches x >= -2?
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Greater than or equal to means at least.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: x is at least -2

4. Solve 2x - 5 < 11. Enter the boundary number.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2x - 5 < 11. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add 5: 2x < 16.
  4. Divide by 2: x < 8.
  5. Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.

Answer: 8

5. Solve -4x >= 20. Enter the boundary number.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve -4x >= 20. Enter the boundary number.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Divide by -4 and flip the sign.
  4. x <= -5, so the boundary is -5.
  5. Check the result by substituting or estimating: the response should match -5 and make sense in the original problem.

Answer: -5

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