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Compound Inequalities

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

A compound inequality joins two inequalities with 'and' or 'or'. An 'and' (like 2 < x < 7) requires both conditions; the solution is the overlap. An 'or' (like x < -3 or x > 5) requires either; the solution is the union. Solve each piece the same way you solve a single inequality.

What you'll learn

Why it matters: Acceptable temperature ranges, age requirements, voltage limits, and quality bands are all compound inequalities — typically 'and' for ranges and 'or' for disqualifying extremes.

Worked example

Problem. Solve 1 < 2x + 3 <= 7.

  1. Subtract 3 from all three parts: -2 < 2x <= 4.
  2. Divide all three parts by 2: -1 < x <= 2.

Answer: -1 < x <= 2

Practice problems

1. In 2 < x < 7, what is the left boundary?

Show solution
  1. Warm-up: First identify exactly what the question is asking: In 2 < x < 7, what is the left boundary?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. The left boundary is the lower bound.
  4. x is greater than 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. For x < -3 or x > 5, give the smallest positive boundary on the right side.

Show solution
  1. Warm-up: First identify exactly what the question is asking: For x < -3 or x > 5, give the smallest positive boundary on the right side.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. The right-hand piece is x > 5.
  4. Its 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

3. Solve 4 <= x + 2 <= 9. Enter the lower bound of x.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 4 <= x + 2 <= 9. Enter the lower bound of x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 2 from all parts: 2 <= x <= 7.
  4. Lower bound is 2.
  5. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

4. Solve 3x > 6 and 2x < 14 (the 'and' overlap). Enter the upper bound of x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 3x > 6 and 2x < 14 (the 'and' overlap). Enter the upper bound of x.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. First: x > 2. Second: x < 7.
  4. Overlap is 2 < x < 7. Upper bound is 7.
  5. Check the result by substituting or estimating: the response should match 7 and make sense in the original problem.

Answer: 7

5. Solve -2 < 3x + 1 < 10. Enter the upper bound of x.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve -2 < 3x + 1 < 10. Enter the upper bound of x.
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Subtract 1: -3 < 3x < 9.
  4. Divide by 3: -1 < x < 3. Upper bound is 3.
  5. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

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