Equations with brackets · Martinとコード

Expand first, then solve. How to deal with brackets and a minus in front of them.

Unwrap first, then solve

The equation 3(x + 2) = 21. Inside the brackets sits x and you need to get to it.

The plan is simple: first unwrap the brackets by expanding — you already can from lesson 11 — and then solve the equation like any other.

3(x + 2) = 21 → I expand: 3x + 6 = 21 → I subtract 6: 3x = 15 → I divide: x = 5.

For this particular equation a shortcut would work too (divide both sides by three right at the start), but expanding works always — even when there are more brackets or they are on both sides. That is why it is taught as the main path.

Today’s only enemy is an old friend: a minus in front of the brackets. It flips the signs of everything inside and anyone who forgets that makes a mistake the check will then mercilessly catch. That is exactly why we never skip the check.

Jonáš and Ema are ordering pizza for the family. Each pizza costs 189 Kč and delivery is 60 Kč. They paid 627 Kč. “How many pizzas did we order?” Ema writes: 189p + 60 = 627. She subtracts delivery: 189p = 567, divides: p = 3. Jonáš adds a harder one: “And if there was 10 Kč off each pizza with delivery included? That is (189 − 10) · p…” Ema nods: “Brackets. First you calculate them or expand them, then you solve on. The order is the whole trick.”

💡

Write the expanded equation all over again on a new line. Anyone who expands “in their head across the line” loses signs. Paper is cheaper than a mistake.

Method for equations with brackets

1. Expand all brackets. The term in front of the brackets times every term inside: 2(3x − 4) = 6x − 8. Arrows help here too.

2. Collect like terms on each side separately. Do not move across the equals sign yet — first tidy each side.

3. Solve like an ordinary equation: x-terms to one side, numbers to the other, divide by the coefficient.

4. Check in the original equation — with the brackets! Substitute and calculate in the order of operations.

A minus in front of brackets: 10 − (x + 3) = 5. The minus flips both terms: 10 − x − 3 = 5 → 7 − x = 5 → x = 2. Check: 10 − (2 + 3) = 10 − 5 = 5. It fits.

Brackets on both sides: 4(x + 1) = 2(x + 5). Expand both: 4x + 4 = 2x + 10. Subtract 2x: 2x + 4 = 10. Subtract 4: 2x = 6. Divide: x = 3.

The order of steps never changes: unwrap → tidy the sides → move → divide → check.

Example 1: 5(x − 2) = 30

Step 1 — I expand: 5x − 10 = 30.

Step 2 — I add 10: 5x = 40.

Step 3 — I divide by five: x = 8.

Check in the original equation: 5 · (8 − 2) = 5 · 6 = 30. It fits.

Notice: in the check I calculate the brackets first (8 − 2 = 6), then I multiply. The check runs in the order of operations, not in the order of my rearrangements.

Example 2: 12 − (2x − 4) = 8

Minus in front of the brackets — I flip both signs inside:

12 − 2x + 4 = 8.

I collect the numbers on the left: 16 − 2x = 8.

I subtract 16: −2x = −8. I divide by −2: x = 4.

Check: 12 − (2 · 4 − 4) = 12 − (8 − 4) = 12 − 4 = 8. It fits.

That +4 after flipping is where half the class falls. A minus in front of brackets changes the minus inside into a plus too.

Example 3: 3(x + 4) = 2(x + 9)

I expand both sides:

3x + 12 = 2x + 18.

I subtract 2x from both sides: x + 12 = 18.

I subtract 12: x = 6.

Check: left 3 · (6 + 4) = 3 · 10 = 30, right 2 · (6 + 9) = 2 · 15 = 30. It fits.

Two pairs of brackets are not a double problem — just twice the same work. Unwrap both and then you go as before.

⚠️

A minus in front of brackets flips the sign of EVERY term inside: −(2x − 4) = −2x + 4, not −2x − 4. One forgotten sign = the whole equation wrong. Luckily the check always catches it.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. スタート with ten. When it goes well, add more.

Solve the equation with brackets and write only x. On paper expand first — train the whole method.

Practise the questions

On paper

Unwrap, tidy, move, divide, check. Five steps on every line of the notebook.

Solve: 4(x + 3) = 32, 6(x − 1) = 24. With expanding on its own line and a check.
Solve an equation with a minus in front of brackets: 20 − (x + 7) = 9.
Solve with brackets on both sides: 2(x + 5) = 3(x + 1).
A ticket in advance is 30 Kč cheaper than on the door. Five tickets in advance cost 600 Kč. Write the equation 5(x − 30) = 600, solve it and write how much a ticket costs on the door.

Now you 💪

I expand brackets first, and only then I move.
A minus in front of brackets flips all the signs inside.
I do the check in the original equation with the brackets.

✅

Done when: You have four solved equations with brackets including a word problem, all with a check, and a run of five correct in practice.

What to take from this lesson

An equation with brackets: expand first, then solve as before.
A minus in front of brackets flips every sign inside.
Tidy each side separately, only then move across the equals sign.
The check belongs in the original equation and runs in the order of operations.
More brackets = more of the same work, not a harder task.