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One-variable inequalities

Solve inequalities, reversing the sign when you divide by a negative.

Practice + answer key · 2 pages

Worksheet settings12 questions · 2 columns · A4
Boundary values

Choose the boundary of the solution set; coefficients can be positive or negative.

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One-variable inequalitiesOne-variable inequalities. Practice. Solve for x: 3x ≤ −3. Solve for x: −6x > 42. Solve for x: −4x > −24. Solve for x: 3x > −30. Solve for x: −x ≤ −6. Solve for x: 7x > 70. Solve for x: −5x ≤ 35. Solve for x: 4x > 16. Solve for x: 8x < −8. Solve for x: 3x ≥ 18. Solve for x: 7x ≥ −35. Solve for x: −2x ≤ 6.Name __________________________Date _______________One-variable inequalitiesSolve each inequality for x.Worked exampleSolve −3x ≤ 6. → x ≥ −2Divide both sides by −3.1.Solve for x: 3x ≤ −3.2.Solve for x: −6x > 42.3.Solve for x: −4x > −24.4.Solve for x: 3x > −30.5.Solve for x: −x ≤ −6.6.Solve for x: 7x > 70.7.Solve for x: −5x ≤ 35.8.Solve for x: 4x > 16.9.Solve for x: 8x < −8.10.Solve for x: 3x ≥ 18.11.Solve for x: 7x ≥ −35.12.Solve for x: −2x ≤ 6.MathSprig · mathsprig.site · support@mathsprig.sitePage 1 of 1mathsprig-1 · mathsprig-v1 · 7a48dd8f
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Practice + answer key · 2 pages

Algebra

What you’ll practice

Solve ax ≤ b, ax < b, ax ≥ b, or ax > b for nonzero a.

An inequality can have many solutions. Isolate x as you would in an equation, but remember to reverse the inequality sign when dividing by a negative number.

How this skill works

Divide both sides by the coefficient of x.

Keep the inequality direction when the coefficient is positive. Reverse it when the coefficient is negative. The boundary can be checked separately from whether it is included.

Worked example

Solve −3x ≤ 6.
  1. Divide both sides by −3.
  2. Reverse ≤ to ≥ because the divisor is negative.
  3. x ≥ 6 ÷ (−3), so x ≥ −2.

Answer: x ≥ −2

Teaching notes

  • An inequality answer is a range of values, not just the boundary number.
Practice questions
  1. Solve for x: 3x ≤ −3.
  2. Solve for x: −6x > 42.
  3. Solve for x: −4x > −24.
  4. Solve for x: 3x > −30.
  5. Solve for x: −x ≤ −6.
  6. Solve for x: 7x > 70.
  7. Solve for x: −5x ≤ 35.
  8. Solve for x: 4x > 16.
  9. Solve for x: 8x < −8.
  10. Solve for x: 3x ≥ 18.
  11. Solve for x: 7x ≥ −35.
  12. Solve for x: −2x ≤ 6.
Answers and steps
  1. x ≤ −1
    1. Divide both sides by 3.
    2. The divisor is positive, so keep ≤.
    3. x ≤ −1.
  2. x < −7
    1. Divide both sides by −6.
    2. The divisor is negative, so reverse > to <.
    3. x < −7.
  3. x < 6
    1. Divide both sides by −4.
    2. The divisor is negative, so reverse > to <.
    3. x < 6.
  4. x > −10
    1. Divide both sides by 3.
    2. The divisor is positive, so keep >.
    3. x > −10.
  5. x ≥ 6
    1. Divide both sides by −1.
    2. The divisor is negative, so reverse ≤ to ≥.
    3. x ≥ 6.
  6. x > 10
    1. Divide both sides by 7.
    2. The divisor is positive, so keep >.
    3. x > 10.
  7. x ≥ −7
    1. Divide both sides by −5.
    2. The divisor is negative, so reverse ≤ to ≥.
    3. x ≥ −7.
  8. x > 4
    1. Divide both sides by 4.
    2. The divisor is positive, so keep >.
    3. x > 4.
  9. x < −1
    1. Divide both sides by 8.
    2. The divisor is positive, so keep <.
    3. x < −1.
  10. x ≥ 6
    1. Divide both sides by 3.
    2. The divisor is positive, so keep ≥.
    3. x ≥ 6.
  11. x ≥ −5
    1. Divide both sides by 7.
    2. The divisor is positive, so keep ≥.
    3. x ≥ −5.
  12. x ≥ −3
    1. Divide both sides by −2.
    2. The divisor is negative, so reverse ≤ to ≥.
    3. x ≥ −3.