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Solving inequalities

We solve an inequality like an equation: the same thing on both sides until x is on its own. If we multiply or divide by a negative number, the inequality sign is reversed. The answer is all the numbers that fit the condition, for example x < 4.

Year 9 and upper secondary · Updated

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What is an inequality?

An inequality says that one side is greater or less than the other. The pointed end of the sign points to the smaller side.

SignRead asTrue examples
<is less than2 < 5
>is greater than5 > 2
≤is less than or equal to4 ≤ 4 and 3 ≤ 4
≥is greater than or equal to4 ≥ 4 and 6 ≥ 4

x > 3 means all numbers greater than 3. With x ≥ 3, the number 3 is included too.

Why the sign is reversed

We want x on its own, exactly as in an equation.

Adding or subtracting the same number does not change the order. Nor does multiplying or dividing by a positive number. But if we multiply by −1, the numbers swap places.

2 < 5, but −2 > −5

−2 degrees is warmer than −5 degrees. That is why we reverse the sign when we multiply or divide by a negative number.

How we check the answer

  • A number that should work: the inequality should be true.
  • A number that should not work: the inequality should not be true.
  • The boundary itself: the sides are equal, and the sign decides whether the boundary is included.

Example

Example 1: 3x + 2 < 14

We want x on its own, with the same steps as in an equation.

  1. We subtract 2 from both sides. The sign stays the same.

    3x + 2 − 2 < 14 − 2

  2. That leaves 3x on the left and 12 on the right.

    3x < 12

  3. We divide both sides by 3. Three is positive, so the sign stays the same.

    3x ÷ 3 < 12 ÷ 3

  4. Now x is on its own. All numbers less than 4 work.

    x < 4

x = 3 gives 11 < 14, which is true. x = 5 gives 17 < 14, which is not true. x = 4 gives 14 < 14, so 4 is not included.

Example 2: 5 − 2x ≥ 11

We want x on its own. In front of x is −2, so the sign is reversed when we divide.

  1. We subtract 5 from both sides. That does not reverse the sign.

    5 − 2x − 5 ≥ 11 − 5

  2. That leaves −2x on the left and 6 on the right.

    −2x ≥ 6

  3. We divide both sides by −2. The number is negative, so ≥ is reversed to ≤.

    −2x ÷ (−2) ≤ 6 ÷ (−2)

  4. All numbers that are −3 or less work.

    x ≤ −3

  5. Another way: add 2x and subtract 11 on both sides. Then −6 ≥ 2x, which gives the same answer.

    −3 ≥ x

x = −4 gives 13 ≥ 11, which is true. x = 0 gives 5 ≥ 11, which is not true. x = −3 gives 11 ≥ 11, so −3 is included.

Example 3: how far will the money go?

The taxi costs 40 kr plus 15 kr/km. We have 250 kr. How many kilometres, x, can we travel?

  1. The journey costs 40 + 15x kr and may cost at most 250 kr.

    40 + 15x ≤ 250

  2. We subtract 40 from both sides.

    15x ≤ 210

  3. We divide both sides by 15. The number is positive, so the sign stays the same.

    x ≤ 14

14 km costs 40 + 15 · 14 = 250 kr. 15 km costs 265 kr. We can travel at most 14 km.

Pitfalls

  • Forgetting to reverse the sign when dividing by a negative number, so that −2x ≥ 6 becomes x ≥ −3.

    Ask yourself: am I dividing by a negative number? Then the sign is reversed. Try x = 0 in 5 − 2x ≥ 11: 5 ≥ 11 is not true.

  • Reversing the sign when subtracting, so that x + 5 < 2 becomes x > −3.

    The sign is reversed only when we multiply or divide by a negative number. The correct answer is x < −3.

  • Answering with a single number, like x = 4, or mixing up < and ≤.

    The answer is an inequality, not a single number. The boundary is included with ≤ and ≥, but not with < and >.

Practise on your own

0 of 4 correct

  1. Solve the inequality x + 7 > 12.

  2. Solve the inequality 4x − 3 ≤ 25.

  3. Solve the inequality 6 − 3x < 18.

  4. Solve the inequality 2x + 9 ≥ 5x − 6.

Common questions

What is the difference between < and ≤?

With x < 4, the number 4 is not included. With x ≤ 4, less than or equal to, 4 is included.

How do I draw the answer on a number line?

Draw a circle at the boundary and an arrow in the direction where the answers are. The circle is filled in if the boundary is included (≤, ≥) and empty otherwise (<, >).

Can an inequality have no solution?

Yes. x + 1 < x is never true. If x disappears and what is left is false, the inequality has no solution.

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