Solving Equations with Brackets

Equations with brackets

An equation says two expressions are equal, like 2(x + 3) = 10. Solving means finding the value of x that makes the statement true. When brackets appear, there is one extra first step: expand the brackets, then solve as an ordinary linear equation.

This lesson combines two skills you've already met — expanding brackets and balancing equations from solving linear equations. If either feels shaky, revisit it, because here they work together.

The balance rule

Think of an equation as a set of scales. The two sides are equal, so the scales are level. To keep them level, whatever you do to one side you must do to the other. Add, subtract, multiply, or divide both sides by the same amount and the balance holds.

This rule is how we peel away everything around x until x stands alone.

The four-step method

1. Expand the brackets.
2. Collect like terms and gather variable terms on one side, numbers on the other.
3. Divide to leave a single x.
4. Check by substituting back into the original equation.

Worked example 1 — bracket on one side

Solve 2(x + 3) = 10.

1. Expand: 2(x + 3) = 2x + 6, so the equation is 2x + 6 = 10.
2. Subtract 6 from both sides: 2x + 6 − 6 = 10 − 6, giving 2x = 4.
3. Divide by 2: x = 4 ÷ 2 = 2.
4. Check: 2(2 + 3) = 2(5) = 10. ✓ Matches the right side.

Worked example 2 — subtraction inside the bracket

Solve 3(x − 1) = 12.

1. Expand: 3(x − 1) = 3x − 3, so 3x − 3 = 12.
2. Add 3 to both sides: 3x = 15.
3. Divide by 3: x = 5.
4. Check: 3(5 − 1) = 3(4) = 12. ✓

Notice the bracket contained a minus, so expanding gave −3, and we added 3 to undo it. Always undo with the opposite operation.

Worked example 3 — variable on both sides

Solve 2(x + 1) = x + 5.

When x appears on both sides, gather the x terms together first.

1. Expand the left side: 2x + 2 = x + 5.
2. Subtract x from both sides to collect variables on the left: 2x − x + 2 = 5, which is x + 2 = 5.
3. Subtract 2: x = 3.
4. Check: left side 2(3 + 1) = 2(4) = 8; right side 3 + 5 = 8. ✓ Both equal 8.

Worked example 4 — brackets on both sides

Solve 3(x + 2) = 2(x + 5).

1. Expand both sides: 3x + 6 = 2x + 10.
2. Subtract 2x from both sides: x + 6 = 10.
3. Subtract 6: x = 4.
4. Check: left 3(4 + 2) = 3(6) = 18; right 2(4 + 5) = 2(9) = 18. ✓

When both sides have brackets, expand both before doing anything else, then collect terms as usual.

Worked example 5 — a negative outside the bracket

Solve 4x − 2(x − 1) = 8.

A minus in front of the bracket flips the signs inside when you expand.

1. Expand −2(x − 1) = −2x + 2. The equation becomes 4x − 2x + 2 = 8.
2. Collect like terms on the left: 4x − 2x = 2x, so 2x + 2 = 8.
3. Subtract 2: 2x = 6.
4. Divide by 2: x = 3.
5. Check: 4(3) − 2(3 − 1) = 12 − 2(2) = 12 − 4 = 8. ✓

The most common error here is writing −2x − 2 instead of −2x + 2. Remember: negative times negative is positive.

Activity — solve each equation

1. 2(x + 4) = 14
2. 5(x − 2) = 20
3. 3(x + 1) = 2x + 7
4. 4(x − 3) = 2(x + 1)
5. 6 + 2(x − 1) = 14
6. 3(x + 2) = 5x − 4

Answers: 1) x = 3 2) x = 6 3) x = 4 4) x = 7 5) x = 5 6) x = 5

(For number 6: expand to 3x + 6 = 5x − 4; subtract 3x to get 6 = 2x − 4; add 4 to get 10 = 2x; divide to get x = 5.)

Why this matters

Many real equations come wrapped in brackets — a discount applied to a total, a formula rearranged, a geometry rule. Being able to expand and then balance lets you unlock the unknown in all of them. It is also the gateway to harder equations and eventually solving two-step equations and beyond.

Stick to the routine: expand, collect, divide, check. The expanding handles the brackets, the balance rule keeps the equation true, and the final substitution proves your answer right.