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Inequalities and absolute value

Solve inequalities like 2x + 1 > 7, see why the sign flips when you divide by a negative, and solve |x − 3| = 5.

Many rules in life are not about one exact number. "You must be at least 12 years old to ride." "Hand luggage must weigh less than 8 kg." "Speed limit 50." These are inequalities: statements that one thing is bigger or smaller than another. In maths we solve them almost exactly like equations, with one important twist. The answer is not one number but a whole range of numbers. This page also covers absolute value, which is about distance and often gives two answers.

The four signs

SignRead it asExample
>is greater than7 > 3
<is less than3 < 7
≥is greater than or equal toage ≥ 12
≤is less than or equal tospeed ≤ 50

An answer like x > 3 means: every number bigger than 3 works. So 4 works, 3.5 works, 100 works, but 3 itself does not, because 3 is not bigger than 3. An answer like x ≥ 3 includes 3 as well. On a number line, people draw x > 3 as an empty circle at 3 with an arrow to the right, and x ≥ 3 with a filled circle.

x > 3: every number to the right of 3 works, like 3.5, 4 and 100. The circle at 3 is empty, because 3 itself does not work.
x ≥ 3: the same, but the circle at 3 is filled in, because 3 works too.

Solving an inequality

The plain method is the same as for an equation. You do the same thing to both sides until x is alone. Adding or subtracting a number on both sides never changes the sign. Multiplying or dividing both sides by a positive number does not change it either.

  1. Undo the adding or subtracting: do the opposite to both sides.
  2. Undo the multiplying: divide both sides by the number in front of x.
  3. Keep the same sign the whole way (while you only divide by positive numbers).
  4. Check with one number that should work and one that should not.

Example 1

Solve 2x + 1 > 7.

  1. Take 1 away from both sides: 2x > 6.
  2. Divide both sides by 2: x > 6/2.
  3. So x > 3.
  4. Check x = 4: 2(4) + 1 = 9, and 9 > 7 is true.
  5. Check x = 2: 2(2) + 1 = 5, and 5 > 7 is false. Good.

Answer: x > 3

Example 2

Solve 3x − 5 ≤ 13.

  1. Add 5 to both sides: 3x ≤ 18.
  2. Divide both sides by 3: x ≤ 18/3.
  3. So x ≤ 6.
  4. Check x = 6: 18 − 5 = 13, and 13 ≤ 13 is true.

Answer: x ≤ 6

Why the sign flips for a negative number

Here is the twist. Start with something true: 2 < 5. Now multiply both sides by −1. You get −2 and −5. Which is bigger? On a cold day, −2 degrees is warmer than −5 degrees, so −2 is the bigger number: −2 > −5. The sign had to turn around to stay true.

On the right, 2 is less than 5. On the left, −2 is to the right of −5, so −2 is the bigger number. Multiplying by −1 turns the order around.

Remember

When you multiply or divide both sides by a negative number, turn the sign around: > becomes <, < becomes >, ≤ becomes ≥, and ≥ becomes ≤.

Adding or subtracting a negative number does not flip it. Only multiplying or dividing does.

Example 3

Solve −3x > 12.

  1. Divide both sides by −3. This is a negative number, so > turns into <.
  2. x < 12/(−3).
  3. So x < −4.
  4. Check x = −5: −3(−5) = 15, and 15 > 12 is true.
  5. Check x = 0: −3(0) = 0, and 0 > 12 is false. Good.

Answer: x < −4

Example 4

Solve 7 − x ≤ 3.

  1. Take 7 away from both sides: −x ≤ −4.
  2. Multiply both sides by −1. The sign turns around: x ≥ 4.
  3. Check x = 5: 7 − 5 = 2, and 2 ≤ 3 is true.

Answer: x ≥ 4

Absolute value

The absolute value of a number is its distance from 0 on the number line. It is written with two straight lines: |5|. A distance is never negative, so |5| = 5 and |−5| = 5 too. Both numbers are 5 steps away from 0, one to the right and one to the left.

Both 5 and −5 are 5 steps from 0, so |5| = 5 and |−5| = 5.

In the same way, |x − 3| is the distance between x and 3. So |x − 3| = 5 asks: which numbers are 5 steps away from 3? Count 5 to the right and you reach 8. Count 5 to the left and you reach −2. There are two answers.

Start at 3. Five steps to the right is 8, and five steps to the left is −2.

Remember

If |A| = k and k is positive, then A = k or A = −k.

Example 5

Solve |x − 3| = 5.

  1. What is inside the lines is 5 or −5.
  2. First case: x − 3 = 5, so x = 8.
  3. Second case: x − 3 = −5, so x = −2.
  4. Check: |8 − 3| = |5| = 5 and |−2 − 3| = |−5| = 5.

Answer: x = 8 or x = −2

Example 6

Solve |x + 4| = 6.

  1. First case: x + 4 = 6, so x = 2.
  2. Second case: x + 4 = −6, so x = −10.
  3. Check: |2 + 4| = 6 and |−10 + 4| = |−6| = 6.

Answer: x = 2 or x = −10

Try it yourself

  1. 1.Solve 3x + 4 > 19.

    Show the answer

    x > 5

    Take 4 away: 3x > 15. Divide by 3: x > 5.

  2. 2.Solve 6x − 5 ≤ 13.

    Show the answer

    x ≤ 3

    Add 5: 6x ≤ 18. Divide by 6: x ≤ 3.

  3. 3.Solve −4x < 20.

    Show the answer

    x > −5

    Divide both sides by −4, and < turns into >: x > −5.

  4. 4.Solve 10 − 2x ≥ 4.

    Show the answer

    x ≤ 3

    Take 10 away: −2x ≥ −6. Divide by −2 and turn the sign: x ≤ 3.

  5. 5.Solve |x − 4| = 6.

    Show the answer

    x = 10 or x = −2

    x − 4 = 6 gives 10, and x − 4 = −6 gives −2. Both are 6 steps from 4.

  6. 6.Solve |x + 1| = 3.

    Show the answer

    x = 2 or x = −4

    x + 1 = 3 gives 2, and x + 1 = −3 gives −4.

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