Expanding always works
One way to solve 3(x + 1) = 12 is to expand the bracket first. The 3 multiplies both terms inside, so the equation becomes 3x + 3 = 12. Subtract 3 from both sides: 3x = 9. Divide both sides by 3: x = 3.
That method works on every equation with a bracket. But there is often a shorter way, and it comes from reading the bracket as one quantity.
Three lots of the same thing
3(x + 1) means 3 lots of x + 1. Whatever number x + 1 turns out to be, three of them together make 12. So one of them is 12 ÷ 3 = 4.
Written as a step, divide both sides by 3. The left side becomes x + 1, and the right side becomes 12 ÷ 3 = 4, so x + 1 = 4. The bracket now stands alone, and one more step finishes it: subtract 1 from both sides, and x = 3. That is the same answer as before, in two steps instead of three.
The three equal lots of x + 1 are as long as 12. Cut 12 into three equal pieces and each is 4, so x + 1 = 4.
Take 1 off both pans. x is left on one side and 4 − 1 = 3 on the other, so x = 3.
When dividing first is the better move
Divide first when the number outside the bracket divides the other side exactly. In 4(x − 2) = 20, 4 divides 20, so divide both sides by 4: x − 2 = 5. Then add 2 to both sides: x = 7. Check: 4 × (7 − 2) = 4 × 5 = 20.
When it does not divide exactly, expanding is usually easier. In 3(x + 2) = 14, dividing by 3 gives , a fraction. Expanding gives 3x + 6 = 14, then 3x = 8 and , with no fraction until the last step.
The question may want the bracket itself
Sometimes the bracket is exactly what the question asks for. If 5(2a − b) = 40, what is 2a − b? Divide both sides by 5. The left side becomes 2a − b, and the right side becomes 40 ÷ 5 = 8, so 2a − b = 8.
You cannot find a or b from this equation, because many pairs of numbers work: a = 5 and b = 2, or a = 4 and b = 0, and more. But every one of those pairs gives 2a − b = 8. The question asked only for 2a − b, so neither letter is needed.
Spotting a multiple of the expression
The same idea works in the other direction. If 2x − 5 = 7, what is 4x − 10?
Look at how the two expressions are related. Double every term of 2x − 5: 2 × 2x = 4x and 2 × (−5) = −10. So 4x − 10 = 2(2x − 5). Since 2x − 5 is 7, 4x − 10 is 2 × 7 = 14.
You could also solve for x first: 2x = 12, so x = 6, and then 4 × 6 − 10 = 14. The answer is the same, but finding x was never needed.
The usual mistakes
Dividing one side only. In 3(x + 1) = 12, dividing the left side by 3 and leaving the 12 gives x + 1 = 12, which is false. Both sides are divided by 3, so x + 1 = 4.
Solving for the letters when the question asks for an expression. If the question asks for 2a − b, the answer is a number for the whole expression, 8, not a value of a or of b.