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Grades 6 to 9 (Years 7 to 11 in England)

Solving inequalities: one step, two step and compound

Here is how to solve inequalities the same way you solve equations, with the one rule that is different: flip the sign when you multiply or divide by a negative number. Try the free solving inequalities practice below, then print a fresh worksheet with an answer key.

The method, step by step

  1. Treat the inequality like an equation and undo the operations on x in reverse order.
  2. Do the same thing to both sides each time.
  3. If you multiply or divide both sides by a negative number, flip the inequality sign.
  4. Write the answer with x on the left, such as x > 4.
  5. Check with a number from your answer, then graph the solution on a number line.

CCSS 6.EE.B.5CCSS 6.EE.B.8CCSS 7.EE.B.4.BCCSS HSA.REI.B.3England: KS3 AlgebraEngland: GCSE Foundation AlgebraEngland: GCSE Higher Algebra

Worked example

Solve 5 − 3x ≤ 17.

  1. Subtract 5 from both sides: −3x ≤ 12.
  2. Divide both sides by −3. Dividing by a negative flips the sign, so ≤ becomes ≥.
  3. 12 ÷ (−3) = −4, so x ≥ −4.
  4. Check x = 0, which is in the answer: 5 − 3 × 0 = 5, and 5 ≤ 17 is true.
  5. Check x = −5, which is not: 5 − 3 × (−5) = 20, and 20 ≤ 17 is false.
  6. x ≥ −4.

Practice now

Question 1 of 100 correct

Solve: 4x − 3 > 5

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What an inequality means

An inequality compares two amounts that may not be equal. The four signs are < (less than), > (greater than), ≤ (less than or equal to) and ≥ (greater than or equal to). An equation like x + 2 = 7 has one answer, but an inequality like x + 2 < 7 has lots: every number less than 5 works, including 4, 0, −10 and 4.9.

Words in story problems match the signs. At least 18 means x ≥ 18, at most 30 means x ≤ 30, more than 12 means x > 12 and fewer than 6 means x < 6. At least and at most include the number itself; more than and less than do not.

Graphing inequalities on a number line

Show the answer on a number line with a circle and an arrow. Use an open circle for < and >, because the number itself is not included. Use a closed, filled-in circle for ≤ and ≥, because it is. Then shade toward the solutions.

  • x > 3: open circle at 3, shade to the right.
  • x ≤ −2: closed circle at −2, shade to the left.
  • −1 < x ≤ 4: open circle at −1, closed circle at 4, shade between them.

Solving one step inequalities

Use inverse operations, exactly as with equations. Adding or subtracting never changes the sign, and neither does multiplying or dividing by a positive number.

  • x + 8 > 15: subtract 8, so x > 7.
  • x − 4 ≤ 9: add 4, so x ≤ 13.
  • 6x ≥ 42: divide by 6, so x ≥ 7.
  • x ÷ 3 < 5: multiply by 3, so x < 15.

Why the sign flips with negatives

Multiplying or dividing by a negative number reverses the order of numbers. Start with a true statement, 2 < 5. Multiply both sides by −1 and you get −2 and −5, but −2 is greater than −5. To stay true, the sign must flip: −2 > −5.

The same happens when you solve. In −4x < 20, divide both sides by −4 and flip the sign: x > −5. Test x = 0: −4 × 0 = 0, and 0 < 20 is true. If you forget to flip, you get x < −5, and testing x = −6 gives 24 < 20, which is false.

Two step inequalities

Undo adding or subtracting first, then multiplying or dividing, just as in two step equations. Flip the sign only at the step where you multiply or divide by a negative.

  • 3x + 7 > 22: subtract 7 to get 3x > 15, then divide by 3, so x > 5.
  • x/2 − 6 ≤ 1: add 6 to get x/2 ≤ 7, then multiply by 2, so x ≤ 14.
  • 10 − 2x ≥ 4: subtract 10 to get −2x ≥ −6, then divide by −2 and flip, so x ≤ 3.
  • A gym charges $20 to join plus $8 a visit, and you can spend at most $100. Then 20 + 8v ≤ 100, so 8v ≤ 80 and v ≤ 10: up to 10 visits.

Compound inequalities

A compound inequality puts two conditions together, such as −3 < 2x + 1 ≤ 9. Do the same thing to all three parts. Subtract 1 to get −4 < 2x ≤ 8, then divide by 2 to get −2 < x ≤ 4. The integers that fit are −1, 0, 1, 2, 3 and 4, which makes six.

How it is taught in school

In the US, Common Core starts in grade 6, where students write inequalities such as x > c to describe a condition, see that they have infinitely many solutions, and show them on a number line (6.EE.B.8). Grade 7 solves word problems leading to inequalities like px + q > r and graphs the solution set (7.EE.B.4.b). High school algebra solves linear inequalities in one variable in general (HSA-REI.B.3), and compound inequalities are usually taught in Algebra 1.

In England, pupils use the symbols <, >, ≤ and ≥ in Key Stage 3. Solving linear inequalities in one variable and showing the solution set on a number line is part of GCSE Mathematics, and Higher tier pupils also write solutions in set notation and show them on graphs.

How parents can help

  • Make sure solving equations feels comfortable first. Solving an inequality is the same process with one extra rule.
  • Use everyday limits. You need at least 8 hours of sleep is h ≥ 8, and a ride for children under 12 is a < 12.
  • Ask your child to test one number from their answer in the original inequality. If it doesn't work, something went wrong.
  • Practice the sign flip with plain numbers first: 3 < 7, so how do −3 and −7 compare?
  • Keep sessions short. Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.

Common mistakes

  • Forgetting to flip the sign when dividing by a negative. −2x > 10 gives x < −5, not x > −5.
  • Flipping the sign when you shouldn't. Subtracting, or dividing by a positive number, never changes the sign: x − 6 < −2 gives x < 4.
  • Flipping because the answer is negative. 3x < −12 gives x < −4; you divided by 3, which is positive, so the sign stays.
  • Using the wrong circle. x > 2 needs an open circle at 2, because 2 itself is not a solution. x ≥ 2 needs a closed one.
  • Shading the wrong way when x is on the right. 7 > x means x < 7, so shade to the left of 7.
  • Changing only two parts of a compound inequality. In 1 < x + 3 < 8, subtract 3 from all three parts: −2 < x < 5.

Questions people ask

How do you solve inequalities?
The same way as equations, by doing the same thing to both sides. The only extra rule is to flip the sign when you multiply or divide by a negative. For 2x + 3 < 11, subtract 3 and divide by 2, so x < 4.
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number. −5x ≤ 30 becomes x ≥ −6.
What is the difference between an open and a closed circle?
An open circle means the number is not included, for < and >. A closed circle means it is included, for ≤ and ≥.
What is a compound inequality?
Two inequalities joined together, such as 2 < x ≤ 7, which means x is greater than 2 and at most 7. Solve it by doing the same step to all three parts.
How do I check the answer to an inequality?
Pick one number inside your answer and one outside it, and test both in the original inequality. The first should make it true and the second false.
What grade learns solving inequalities?
In the US, grade 6 introduces inequalities on a number line (6.EE.B.8), grade 7 solves two step inequalities (7.EE.B.4.b), and Algebra 1 adds compound inequalities. In England, solving linear inequalities is part of GCSE Mathematics.
Is there a free solving inequalities worksheet with answers?
Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

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