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Equations with Brackets and Fractions

Clear brackets and fractions first, then solve the simpler equation left behind.

Clearing Brackets First

If an equation has brackets, expand them first. This turns the equation into the simple form you already know how to solve.

Example 1 — a bracket

Solve 2 (x + 3) = 16 . 1 Expand the bracket: 2 x + 6 = 16 2 Subtract 6 from both sides: 2 x = 10 3 Divide both sides by 2 : x = 5 Check: 2 (5 + 3) = 2 (8) = 16 ✓ Answer: x = 5

Example 2 — brackets on both sides

Solve 3 (x - 1) = 2 (x + 4). 1 Expand both sides: 3 x - 3 = 2 x + 8 2 Subtract 2 x from both sides: x - 3 = 8 3 Add 3 to both sides: x = 11 Check: 3 (11 - 1) = 30, and 2 (11 + 4) = 30 ✓ Answer: x = 11

If an equation has a fraction, multiply every term on both sides by the denominator. This removes the fraction completely.

Example 1 — a simple fraction

Solve \frac{x}{3} + 2 = 7 . 1 Subtract 2 from both sides: \frac{x}{3} = 5 2 Multiply both sides by 3 : x = 15 Check: \frac{15}{3} + 2 = 5 + 2 = 7 ✓ Answer: x = 15

Example 2 — fraction with the whole side

Solve (x + 4) \div 2 = 9 . 1 Multiply both sides by 2 : x + 4 = 18 2 Subtract 4 from both sides: x = 14 Check: (14 + 4) \div 2 = \frac{18}{2} = 9 ✓ Answer: x = 14

Practice

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1 / 2

Solve 3 (x + 2) = 21 .

Solve \frac{x}{4} - 1 = 3 . What is x ?

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