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1
What You'll Master
objectives
Know
Brackets mean multiply the outside term by every term inside — that's the distributive law
The distributive law: $a(b + c) = ab + ac$
Understand
Two strategies exist: expand-then-solve and divide-first
Strategy B (divide first) only works when the bracket stands alone — no extra terms outside
Can Do
Solve equations like $3(x + 4) = 21$ using either strategy
Choose the most efficient strategy for a given equation
2
Words You Need
vocabulary
ExpandMultiply the term outside the brackets by every term inside. $3(x + 4)$ expands to $3x + 12$.
Distributive Law$a(b + c) = ab + ac$. The outside factor distributes over (multiplies) every term inside the brackets.
Brackets EquationAn equation where the unknown is inside brackets, like $5(x - 3) = 25$. Requires an extra step to solve.
Collect Like TermsCombine terms that have the same variable (or no variable). Used after expanding to simplify.
Factor OutThe reverse of expanding — write $3x + 12$ back as $3(x + 4)$. Useful for checking your work.
3
Strategy A: Expand Then Solve
+5 XP
Works for ALL brackets equations. Expand the brackets first using the distributive law, then solve the resulting two-step equation.
To expand $3(x + 4)$, multiply the 3 by every term inside the brackets: $3 \times x = 3x$ and $3 \times 4 = 12$. So $3(x + 4) = 3x + 12$. Then solve as usual.
$3(x+4)=21 \;\Rightarrow\; 3x+12=21$
Step 1: Expand
$3(x + 4) = 21 \;\Rightarrow\; 3x + 12 = 21$
Step 2: Subtract
$3x + 12 - 12 = 21 - 12 \;\Rightarrow\; 3x = 9$
Step 3: Divide & Check
$x = 3$. Check: $3(3+4) = 3(7) = 21$ ✓
4
Strategy B: Divide First (When Possible)
+5 XP
Faster when it applies — skip the expanding step entirely by dividing both sides by the coefficient first. Only works when the bracket stands alone (no extra terms outside).
Step 1: Divide both sides by 3
$3(x + 4) = 21 \;\Rightarrow\; x + 4 = 7$
Step 2: Subtract 4
$x + 4 - 4 = 7 - 4 \;\Rightarrow\; x = 3$ ✓
Warning: Cannot divide first here
$3(x+4) + 5 = 26$ — the $+5$ outside the bracket blocks Strategy B. Use Strategy A instead.
OK for Strategy B: $3(x + 4) = 21$ — bracket stands alone on the left. Divide both sides by 3.
Must use Strategy A: $3(x + 4) + 5 = 26$ — the $+5$ outside means you cannot divide first. Expand instead.
5
Comparing Both Strategies
+5 XP
Solve $4(2x - 3) = 20$ using both strategies side by side. Both give $x = \frac{7}{2} = 3.5$.
Recommendation: Try Strategy B first. If the right-hand side divides evenly by the coefficient outside the brackets and there are no extra terms outside, it's faster. Otherwise, use Strategy A — it always works.
Watch Me Solve It · Worked example
Watch Me Solve It · Solve $2(3x + 1) = 20$
+15 XP per step
Q
PROBLEM
Solve $2(3x + 1) = 20$ using Strategy A (expand first). Show all working and check your answer.
1
Expand the brackets using the distributive law
$2(3x + 1) = 20 \;\Rightarrow\; 6x + 2 = 20$
Multiply the 2 by every term inside: $2 \times 3x = 6x$ and $2 \times 1 = 2$. Write the expanded equation.
2
Subtract 2 from both sides
$6x + 2 - 2 = 20 - 2 \;\Rightarrow\; 6x = 18$
Do the same thing to both sides to keep the equation balanced. Subtracting 2 isolates the $6x$ term.
3
Divide both sides by 6
$6x \div 6 = 18 \div 6 \;\Rightarrow\; x = 3$
Dividing both sides by the coefficient of $x$ gives the value of $x$.
4
Check by substituting $x = 3$ back into the original equation
$2(3 \times 3 + 1) = 2(9 + 1) = 2(10) = 20$ ✓
Always substitute back into the original equation (with brackets) to verify. Both sides equal 20, so $x = 3$ is correct.
Nice work — XP earned
Answer$x = 3$ · Check: $2(10) = 20$ ✓
6
Common Pitfalls
heads-up
Expanding incorrectly: $3(x + 4) \neq 3x + 4$
A very common error — students only multiply the first term inside the brackets, forgetting the second. The 3 must multiply both $x$ and $4$.
Fix: use an arrow from the outside number to every term inside. $3(x + 4) = 3 \times x + 3 \times 4 = 3x + 12$.
Using Strategy B when there are extra terms outside the bracket
If the equation is $3(x + 4) + 5 = 26$, you cannot divide both sides by 3 first because the $+5$ is not inside the bracket. Dividing would give $\frac{3(x+4)+5}{3}$, which is messy.
Fix: check if the bracket stands completely alone on its side of the equation. If there are any extra terms, expand first (Strategy A).
Forgetting to check by substituting back
A small arithmetic slip anywhere in the working will give a wrong answer. Without checking, you won't know.
Fix: always substitute your answer back into the original equation (with the brackets still in). If both sides match, you're correct.
Quick Check · 5 questions
1
Solve $4(x + 2) = 24$.
+10 XP
2
Expand and solve: $2(x - 5) = 12$.
+10 XP
3
Solve $\frac{x + 3}{4} = 5$.
+10 XP
4
Solve $3(x + 4) = 2(x + 9)$.
+10 XP
5
"Three groups of $(x + 7)$ apples equals 36 apples." Which finds $x$?
+10 XP
ApplyEasy2 MARKS
Q1. Solve $5(x - 3) = 25$ using both methods (expand first AND divide first). Show all working.
Answer in your workbook.
ApplyMedium3 MARKS
Q2. Solve $\frac{x - 4}{3} = 6$. Show your working and check your answer.
Answer in your workbook.
ApplyHard4 MARKS
Q3. A teacher has 6 identical packets of pencils. Each packet contains the same number of pencils plus 2 bonus pencils. In total there are 48 pencils. Write and solve an equation to find how many pencils are in each packet (not counting the bonus).
Answer in your workbook.
Stretch Challenge · +25 XP, +10 coins
Extension Problems
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