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Linear Inequalities

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

A linear inequality compares two expressions instead of setting them equal. Its solution is usually a range of values rather than one number. You solve linear inequalities much like equations, but if you multiply or divide both sides by a negative number, the inequality symbol reverses. This matters because inequalities model limits, budgets, minimum requirements, and intervals. When practicing, isolate the variable, track the direction of the symbol, and interpret the answer as a set of allowed values. A common mistake is forgetting to flip the symbol after dividing by a negative coefficient.

What you'll learn

Why it matters: Spending limits, minimum score requirements, safe temperature ranges, and eligibility cutoffs are inequalities rather than single answers. The number-line view helps students see the boundary and the whole interval of values that work.

Worked example

Problem. Solve -3x + 4 < 19.

  1. Subtract 4: -3x < 15.
  2. Divide by -3 and reverse the sign.
  3. x > -5.

Answer: x > -5

Practice problems

1. Solve x + 9 < 15. Enter the boundary number.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x + 9 < 15. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 9.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

2. Solve 4x >= 28. Enter the boundary number.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 4x >= 28. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Divide by 4.
  4. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

3. Solve -2x <= 10. Enter the boundary number.

Show solution
  1. Challenge: First identify exactly what the question is asking: Solve -2x <= 10. 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 -2 and reverse the inequality.
  4. Check the result by substituting or estimating: the response should match -5 and make sense in the original problem.

Answer: -5

4. Solve 2x - 7 <= 11. Enter the boundary number.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2x - 7 <= 11. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add 7, then divide by 2.
  4. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

5. Solve -5x + 3 > 28. Enter the boundary number.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve -5x + 3 > 28. Enter the boundary number.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Subtract 3, then divide by -5 and reverse the sign.
  4. 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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