Lesson Guide
Explanation
When an equation has brackets, the simplest route is to expand them first and then solve exactly as you already know how. In 3(x + 4) = 27, the 3 outside multiplies everything inside, giving 3x + 12 = 27. From there it is an ordinary two-step equation: subtract 12 to get 3x = 15, then divide by 3 to get x = 5. Check by substituting into the original with the brackets still there: 3 times (5 + 4) is 3 times 9, which is 27 ✓.
Your Turn
Solve 2(x + 3) = 16. Check: expand to 2x + 6 = 16, so 2x = 10 and x = 5. Then 2(5 + 3) = 16 ✓
Watch Out
3(x + 4) is NOT 3x + 4. The multiplier visits every term inside the bracket — this is the single most common error in the chapter.
Explanation
There is a shortcut worth knowing. Since the whole bracket is multiplied by 3, you can divide both sides by 3 straight away: 3(x + 4) = 27 becomes x + 4 = 9, so x = 5 in one further step. That is quicker — but only when the number outside divides the right-hand side exactly. If the equation were 3(x + 4) = 28, dividing would give 9.33 and get ugly, so there you would expand instead. Glance at the right-hand side first: if it divides cleanly, divide; if not, expand.
Your Turn
Solve 5(x − 2) = 35 using the shortcut. Check: divide by 5 → x − 2 = 7, so x = 9.
Watch Out
If you divide first, divide the WHOLE right-hand side, and remember the bracket disappears — do not divide only part of it.
Explanation
Negatives are where brackets bite. A negative outside the bracket flips the sign of every term inside, because a negative times a negative is a positive. So −2(x − 5) is −2x + 10, not −2x − 10. The same care applies with a minus inside: in 2(x − 3) + 4 = 14, expanding gives 2x − 6 + 4, which tidies to 2x − 2 = 14. Add 2, divide by 2, and x = 8. Slow down at every minus sign — that is where the marks go.
Your Turn
Expand −3(x − 4). Check: −3 × x = −3x and −3 × (−4) = +12, so −3x + 12.
Watch Out
A bare minus in front of a bracket means × (−1). So −(x − 7) is −x + 7.
Explanation
With brackets on both sides, expand each side separately first, then gather the x terms on one side and the numbers on the other. From 5(x − 2) = 3(x + 4) you get 5x − 10 = 3x + 12. Subtract 3x from both sides to leave 2x − 10 = 12, add 10 to get 2x = 22, and divide to get x = 11. A useful habit: move the x terms to whichever side keeps the coefficient positive — here subtracting 3x rather than 5x avoids ending up with −2x, and negatives are where slips happen.
Your Turn
Solve 4(x + 1) = 2(x + 7). Check: 4x + 4 = 2x + 14, so 2x = 10 and x = 5.
Watch Out
Expand both sides completely before you start moving terms. Mixing the two operations in one line is how sign errors creep in.
Explanation
The full method, in order. Expand every bracket — or divide, if the number outside goes into the other side exactly. Tidy each side by collecting like terms. Gather the x terms on one side and the numbers on the other. Divide by whatever x is multiplied by. Then substitute back into the original, brackets and all. Three traps account for nearly every lost mark: forgetting the multiplier visits every term, mishandling a negative outside the bracket, and forgetting that a bare minus means times negative one.
Your Turn
Solve 3(2x − 1) = 2(x + 6) + 1. Check: 6x − 3 = 2x + 13, so 4x = 16 and x = 4.
Watch Out
Substitute into the original equation with the brackets, not into your expanded version — otherwise an expansion error goes undetected.
📌 Key Points — this lesson in a nutshell
- Expand the bracket first, then solve as an ordinary equation.
- The multiplier visits EVERY term inside: 3(x + 4) = 3x + 12, never 3x + 4.
- Shortcut: if the number outside divides the other side exactly, divide first — 3(x + 4) = 27 → x + 4 = 9.
- A negative outside flips the inside signs: −2(x − 5) = −2x + 10.
- A bare minus in front of a bracket means × (−1): −(x − 7) = −x + 7.
- Brackets both sides: expand each separately, then move x terms to keep the coefficient positive.
- Check by substituting into the ORIGINAL equation, brackets included.
Step 1: Introduction
Step 1 of 5
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