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Inequalities

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

Inequalities compare values that may not be equal. The symbols mean less than, greater than, less than or equal to, and greater than or equal to. When multiplying or dividing both sides by a negative number, the inequality direction reverses.

What you'll learn

Why it matters: Speed limits, age restrictions, dosage ranges, and budget caps are stated as inequalities. The solution is the whole set of values that work, which is why a number-line picture is the natural way to show it.

Worked example

Problem. Solve x + 4 < 10.

  1. Subtract 4 from both sides.
  2. x < 6.
  3. The solution is every number less than 6.

Answer: x < 6

Practice problems

1. Which phrase matches x > 8?

Choices: x is greater than 8 · x is less than 8 · x equals 8 · x is at most 8

Show solution
  1. Warm-up: First identify exactly what the question is asking: Which phrase matches x > 8?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. The symbol > means greater than.
  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 greater than 8

2. Solve x + 3 < 9. Enter the boundary number.

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

Answer: 6

3. Solve y - 5 > 12. Enter the boundary number.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve y - 5 > 12. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Add 5 to both sides.
  4. y > 17, so the boundary number is 17.
  5. Check the result by substituting or estimating: the response should match 17 and make sense in the original problem.

Answer: 17

4. Which numbers satisfy x < 4?

Choices: 3 only · 4 only · 5 only · 4 and 5

Show solution
  1. Core Practice: First identify exactly what the question is asking: Which numbers satisfy x < 4?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Numbers less than 4 satisfy x < 4.
  4. 3 is less than 4.
  5. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: 3 only

5. Solve 2x <= 14. Enter the boundary number.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 2x <= 14. Enter the boundary number.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Divide both sides by 2.
  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

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