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Equations with brackets

First we multiply every term inside the bracket by the number in front of it. Then the bracket is gone and we solve an ordinary equation. If there is a minus in front of the bracket, every term inside changes sign.

Years 7–9 and upper secondary · Updated

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The number in front of the bracket

A number in front of a bracket means multiplication. 3(x + 4) is three groups of x + 4, so three x and three fours.

3(x + 4) = (x + 4) + (x + 4) + (x + 4) = 3x + 12

A minus in front of the bracket

A minus in front takes away the whole bracket. That is why every term inside changes sign. We test it with numbers.

10 − (4 − 1) = 10 − 3 = 7

10 − 4 + 1 = 7 matches, but 10 − 4 − 1 = 5 does not.

−(x − 5) = −x + 5

If there is −2 in front, minus times minus makes plus: −2(x − 3) = −2x + 6.

The method step by step

We want x on its own. First we remove the brackets.

  • Multiply every term inside by the number in front. Watch the signs.
  • Add the x terms together and the numbers together on each side.
  • Collect x on one side by doing the same thing on both sides.
  • Divide by the number in front of x and check the answer.

Example

Example 1: 4(x − 2) = 20

We want x on its own. First we multiply both terms in the bracket by four.

  1. 4 · x = 4x and 4 · (−2) = −8.

    4x − 8 = 20

  2. We add 8 to both sides.

    4x − 8 + 8 = 20 + 8

  3. What is left is four x, which together make 28.

    4x = 28

  4. We divide both sides by 4.

    x = 7

  5. Shortcut: divide both sides by 4 straight away to get x − 2 = 5. Then add 2 to both sides.

    x − 2 = 5, so x = 7

4(7 − 2) = 4 · 5 = 20. The left-hand side is 20, the same as the right. The answer is correct.

Example 2: 3(2x + 1) = 4x + 11

We want x on its own. First we remove the bracket.

  1. 3 · 2x = 6x and 3 · 1 = 3.

    6x + 3 = 4x + 11

  2. We subtract 4x from both sides.

    6x + 3 − 4x = 4x + 11 − 4x

  3. 6x − 4x = 2x, and 11 is left on the right.

    2x + 3 = 11

  4. We subtract 3 from both sides.

    2x = 8

  5. We divide both sides by 2.

    x = 4

Left: 3(2 · 4 + 1) = 3 · 9 = 27. Right: 4 · 4 + 11 = 27. The sides are equal, so x = 4 is correct.

Example 3: a minus in front of the bracket, 20 − 2(x + 3) = 4

We want x on its own. We multiply both terms in the bracket by −2.

  1. −2 · x = −2x and −2 · 3 = −6.

    20 − 2x − 6 = 4

  2. 20 − 6 = 14.

    14 − 2x = 4

  3. We add 2x to both sides, because −2x + 2x = 0.

    14 = 4 + 2x

  4. We subtract 4 from both sides.

    10 = 2x

  5. We divide both sides by 2.

    x = 5

20 − 2(5 + 3) = 20 − 16 = 4. The left-hand side is 4, the same as the right. The answer is correct.

Pitfalls

  • Multiplying only the first term, so that 3(x + 4) becomes 3x + 4.

    The three is multiplied by every term: 3(x + 4) = 3x + 12.

  • Changing the sign of only the first term, so that 12 − (x − 4) becomes 12 − x − 4.

    A minus in front changes the sign of every term: 12 − (x − 4) = 12 − x + 4.

  • Getting the wrong sign with a negative number in front, so that −2(x − 3) becomes −2x − 6.

    (−2) · (−3) = +6, because minus times minus makes plus. The correct answer is −2x + 6.

Practise on your own

0 of 4 correct

  1. Solve the equation 5(x + 2) = 35.

  2. Solve the equation 2(3x − 4) = 16.

  3. Solve the equation 3(x + 5) = 5(x − 1).

  4. Solve the equation 12 − (x − 4) = 2(x + 2).

More walkthroughs with Fox

Common questions

Do you always have to multiply out the bracket first?

No. In 4(x − 2) = 20 you can divide both sides by 4 straight away. Multiplying out the bracket always works.

What does a bracket with no number in front mean?

With a plus in front, you remove the bracket as it is. With a minus in front, every term inside changes sign.

What do I do if two brackets are multiplied together?

Multiply every term in the first by every term in the second. If an x² is left, you solve the equation with the pq formula.

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