Chapter 1 of 8

All 8 chapters
  1. 1Harder linear equations, readunderstood
  2. 2Changing the subject, readunderstood
  3. 3Inequalities, readunderstood
  4. 4Equations of straight lines, readunderstood
  5. 5Two equations: adding or taking away, readunderstood
  6. 6Two equations: substitution, readunderstood
  7. 7Two equations on a graph, readunderstood
  8. 8Word problems and regions, readunderstood

M4.8.1 Chapter 1

Harder linear equations

How do I solve 3(x + 2) = 2(x + 5), with a bracket on both sides?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Solve equations with brackets on both sides
  • Solve equations with fractions, like (x + 2)/3 = 5

Ravi packs boxes. Each box holds one bag of beans and 2 weights of 1 kg. Every bag weighs the same. Call that weight x kg.

Three boxes weigh 3(x + 2) kg. The bracket tells you that the 3 multiplies everything inside it. You get 3 bags and 3 × 2 = 6 weights.

Draw it first

Open the bracket

  1. A rectangle 3 high, labelled 3 on its left side. Its width is x + 2, written over a bracket, with x for the bag and 2 for the weights. Step 2 writes 3x in the first cell and 6 in the second cell. Step 3 writes 3(x + 2) = 3x + 6 under it.x23x + 2

    1 Draw a rectangle 3 high. Make it x + 2 wide: one piece for the bag and one for the 2 weights.

  2. A rectangle 3 high, labelled 3 on its left side. Its width is x + 2, written over a bracket, with x for the bag and 2 for the weights. Step 2 writes 3x in the first cell and 6 in the second cell. Step 3 writes 3(x + 2) = 3x + 6 under it.x23x + 2x23x + 23x6

    2 Each of the 3 rows has an x and a 2. Write 3x for the bags and 6 for the weights.

  3. A rectangle 3 high, labelled 3 on its left side. Its width is x + 2, written over a bracket, with x for the bag and 2 for the weights. Step 2 writes 3x in the first cell and 6 in the second cell. Step 3 writes 3(x + 2) = 3x + 6 under it.x23x + 2x23x + 23x63(x + 2) = 3x + 6

    3 The whole rectangle is the same as its two cells: 3(x + 2) = 3x + 6.

To expand a bracket is to multiply every term inside it by the number outside. A term is one part of an expression, with its sign. In (x + 2) the terms are x and +2.

Put 3 of these boxes on one pan of a balance. On the other pan put 2 boxes that each hold one bag and 5 weights. The pans are level, so 3(x + 2) = 2(x + 5). Open the brackets and look at what is on each pan.

A level balance. The left pan holds 3 bags marked x and 6 weights, and is labelled 3x + 6. The right pan holds 2 bags marked x and 10 weights, and is labelled 2x + 10.xxx3x + 6xx2x + 10
After the brackets are opened: 3x + 6 balances 2x + 10. Each small square is a 1 kg weight.

A balance stays level if you take the same from both pans. Take 2 bags off each pan. The left pan keeps 1 bag and 6 weights. The right pan keeps 10 weights.

A level balance. The left pan holds 1 bag marked x and 6 weights, and is labelled x + 6. The right pan holds 10 weights and is labelled 10.xx + 610
Take 2x from both sides: x + 6 balances 10. Each small square is a 1 kg weight.

Worked example

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

  1. Expand both brackets: 3x + 6 = 2x + 10.
  2. Take 2x from both sides: x + 6 = 10.
  3. Take 6 from both sides: x = 4.
  4. Check: the left side is 3(4 + 2) = 18 and the right side is 2(4 + 5) = 18.

Answer: x = 4

Take care with a minus sign inside a bracket. In 4(x − 1) the 4 multiplies the −1 too, so you get 4x − 4. The minus sign stays with the number.

Worked example

Solve 4(x − 1) = 2(x + 5).

  1. Expand both brackets: 4x − 4 = 2x + 10.
  2. Take 2x from both sides: 2x − 4 = 10.
  3. Add 4 to both sides: 2x = 14.
  4. Divide both sides by 2: x = 7.
  5. Check: the left side is 4(7 − 1) = 24 and the right side is 2(7 + 5) = 24.

Answer: x = 7

Now fractions. The equation (x + 2)/3 = 5 says: x + 2, shared into 3 equal parts, gives 5 in each part.

A bar cut into 3 equal parts. Each part is labelled 5. A bracket over the whole bar is labelled x + 2.555x + 2
Three parts of 5 make x + 2.

So x + 2 must be 3 × 5 = 15. In symbols, multiply both sides of the equation by 3, the number below the line. Every term on a side is multiplied.

Worked example

Solve (x + 2)/3 = 5.

  1. Multiply both sides by 3. On the left, 3 × (x + 2)/3 is x + 2.
  2. On the right, 3 × 5 = 15. So x + 2 = 15.
  3. Take 2 from both sides: x = 13.
  4. Check: (13 + 2)/3 = 15/3 = 5.

Answer: x = 13

See it three ways

With things

Two boxes each hold one bag and 3 weights of 1 kg. Together they weigh 14 kg. Share 14 kg into 2 equal boxes.

As a picture

A bar cut into 2 equal parts. Each part is labelled x + 3 and is 7 long. A bracket over the whole bar is labelled 14 kg.x + 3x + 314 kg

In words

Each box weighs 14 ÷ 2 = 7 kg. A box holds a bag and 3 kg, so the bag weighs 7 − 3 = 4 kg.

In symbols

2(x + 3) = 14. Then x + 3 = 7, so x = 4. Or open the bracket first: 2x + 6 = 14, then 2x = 8, so x = 4.

Try it

Test an answer in both sides

You need: paper, a pencil.

  1. Write 3(x + 2) = 2(x + 5).
  2. Put x = 4 in the left side, then in the right side. Write both answers.
  3. Try x = 5 and x = 3 the same way.

What you will see: Only x = 4 gives the same number on both sides. Any other value leaves one side bigger.

If you cannot try it, this is what happens.
A table with the head x, left, right. Rows: 3, 15, 16; 4, 18, 18; 5, 21, 20.xleftright315164181852120

A wrong idea many people have

4(x − 1) = 4x − 1.

Why it feels right: The 4 sits next to the x, so it feels as if it only touches the x.

The test that breaks it: Put x = 3. Then 4(3 − 1) = 4 × 2 = 8. But 4x − 1 = 12 − 1 = 11. The two do not match, and 4x − 4 = 8 does.

What is true: The number outside multiplies every term inside the bracket, including the one with the minus sign.

A wrong idea many people have

To clear the 3 in (x + 2)/3 = 5, multiply only the x by 3. That gives 3x + 2 = 5.

Why it feels right: The x is the letter you want, so it feels like the only thing to change.

The test that breaks it: From 3x + 2 = 5 you get x = 1. Put x = 1 in the original: (1 + 2)/3 = 1, not 5.

What is true: Multiply the whole of each side by 3. The whole x + 2 is multiplied, not just the x.

Key words

expand
to remove a bracket by multiplying every term inside it by the number outside.
term
one part of an expression, with its sign, such as 3x or −4.

Back to your guess

The answer is C: Multiply every term in each bracket: 3x + 6 = 2x + 10.

The number outside a bracket multiplies every term inside it. So 3(x + 2) is 3x + 6, and 2(x + 5) is 2x + 10. Then you solve 3x + 6 = 2x + 10, and x = 4.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. What is 5(x − 3) when the bracket is opened?

    Tap your answer.

    1. Not this one. You multiplied only the x. The 5 multiplies every term inside, so the 3 is multiplied as well. Go to M3.8.4. Try another.

    2. Not this one. The sign in front of the 3 is a minus, and it stays. 5 × (−3) is −15, not +15. Go to M3.8.4. Try another.

    3. Yes, that is right. 5 × x = 5x and 5 × (−3) = −15. So 5(x − 3) = 5x − 15.

    4. Not this one. You multiplied the 3 by 5 but left the x alone. Both terms inside the bracket are multiplied. Go to M3.8.4. Try another.

    See the answer

    5x − 15

    5 × x = 5x and 5 × (−3) = −15. So 5(x − 3) = 5x − 15.

    • If you chose A: You multiplied only the x. The 5 multiplies every term inside, so the 3 is multiplied as well. Go to M3.8.4.
    • If you chose B: The sign in front of the 3 is a minus, and it stays. 5 × (−3) is −15, not +15. Go to M3.8.4.
    • If you chose D: You multiplied the 3 by 5 but left the x alone. Both terms inside the bracket are multiplied. Go to M3.8.4.

  2. The balance is level. Each small square is a 1 kg weight. How much does one bag weigh?

    A level balance. The left pan holds 2 bags marked x and 7 weights of 1 kg, and is labelled 2x + 7. The right pan holds 1 bag marked x and 12 weights of 1 kg, and is labelled x + 12.xx2x + 7xx + 12
    Tap your answer.

    1. Yes, that is right. Take 7 from both sides: 2x = x + 5. Take x from both sides: x = 5. Check: 2 × 5 + 7 = 17 and 5 + 12 = 17.

    2. Not this one. You added the weights on both sides. To keep a balance level, take the same from both pans instead. Go to M4.8.1. Try another.

    3. Not this one. You took 7 from both pans but left out the bag on the right, so you shared 5 kg between 2 bags. Take one bag off each pan too: 1 bag balances 5 weights. Go to M4.8.1. Try another.

    See the answer

    5 kg

    Take 7 from both sides: 2x = x + 5. Take x from both sides: x = 5. Check: 2 × 5 + 7 = 17 and 5 + 12 = 17.

    • If you chose B: You added the weights on both sides. To keep a balance level, take the same from both pans instead. Go to M4.8.1.
    • If you chose C: You took 7 from both pans but left out the bag on the right, so you shared 5 kg between 2 bags. Take one bag off each pan too: 1 bag balances 5 weights. Go to M4.8.1.

  3. Solve 2(x + 6) = 3(x + 2).

    Tap your answer.

    1. Not this one. You added 12 and 6. After opening the brackets, 2x + 12 = 3x + 6, take 2x and 6 from both sides to get 6 = x. Go to M4.8.1. Try another.

    2. Yes, that is right. Expand: 2x + 12 = 3x + 6. Take 2x from both sides: 12 = x + 6. Take 6 from both sides: 6 = x.

    3. Not this one. You used 6 − 12, the right pair of numbers in the wrong order. Check: x = −6 gives 0 on the left and −12 on the right. Go to M4.8.1. Try another.

    See the answer

    x = 6

    Expand: 2x + 12 = 3x + 6. Take 2x from both sides: 12 = x + 6. Take 6 from both sides: 6 = x.

    • If you chose A: You added 12 and 6. After opening the brackets, 2x + 12 = 3x + 6, take 2x and 6 from both sides to get 6 = x. Go to M4.8.1.
    • If you chose C: You used 6 − 12, the right pair of numbers in the wrong order. Check: x = −6 gives 0 on the left and −12 on the right. Go to M4.8.1.

  4. Solve 3(x − 2) = x + 8.

    Tap your answer.

    1. Not this one. You multiplied the x by 3 but not the −2. The 3 multiplies both terms: 3 × (−2) = −6. Go to M4.8.1. Try another.

    2. Not this one. You turned 3 × (−2) into +6. A minus sign stays a minus sign: 3 × (−2) = −6. Go to M4.8.1. Try another.

    3. Yes, that is right. Expand: 3x − 6 = x + 8. Take x from both sides: 2x − 6 = 8. Add 6: 2x = 14. Divide by 2: x = 7.

    See the answer

    x = 7

    Expand: 3x − 6 = x + 8. Take x from both sides: 2x − 6 = 8. Add 6: 2x = 14. Divide by 2: x = 7.

    • If you chose A: You multiplied the x by 3 but not the −2. The 3 multiplies both terms: 3 × (−2) = −6. Go to M4.8.1.
    • If you chose B: You turned 3 × (−2) into +6. A minus sign stays a minus sign: 3 × (−2) = −6. Go to M4.8.1.

  5. Solve (x + 3)/4 = 3.

    Tap your answer.

    1. Not this one. You solved x + 3 = 3 and left out the 4 below the line. First multiply both sides by 4: x + 3 = 12. Go to M4.8.1. Try another.

    2. Not this one. Multiplying by 4 gives x + 3 = 12. That part is right. Then you must take 3 away, not add it. Go to M4.8.1. Try another.

    3. Not this one. Multiplying by 4 gives x + 3 = 12. The 3 must still be taken from both sides to leave x. Go to M4.8.1. Try another.

    4. Yes, that is right. Multiply both sides by 4: x + 3 = 12. Take 3 from both sides: x = 9. Check: (9 + 3)/4 = 3.

    See the answer

    x = 9

    Multiply both sides by 4: x + 3 = 12. Take 3 from both sides: x = 9. Check: (9 + 3)/4 = 3.

    • If you chose A: You solved x + 3 = 3 and left out the 4 below the line. First multiply both sides by 4: x + 3 = 12. Go to M4.8.1.
    • If you chose B: Multiplying by 4 gives x + 3 = 12. That part is right. Then you must take 3 away, not add it. Go to M4.8.1.
    • If you chose C: Multiplying by 4 gives x + 3 = 12. The 3 must still be taken from both sides to leave x. Go to M4.8.1.

  6. The bar shows (x − 2)/3 = 4. What is x?

    A bar cut into 3 equal parts. Each part is labelled 4. A bracket over the whole bar is labelled x − 2.444x − 2
    Tap your answer.

    1. Yes, that is right. The bar has 3 parts of 4, so x − 2 = 12. Add 2 to both sides: x = 14.

    2. Not this one. You added 2 to the 4. The whole bar is 3 parts of 4, which is 12. Then x − 2 = 12. Go to M4.8.1. Try another.

    3. Not this one. The whole bar is 3 × 4 = 12, so x − 2 = 12. You took 2 away again, but the 2 must be added back. Go to M4.8.1. Try another.

    4. Not this one. The whole bar, 12, is x − 2, not x. Add the 2 back to find x. Go to M4.8.1. Try another.

    See the answer

    x = 14

    The bar has 3 parts of 4, so x − 2 = 12. Add 2 to both sides: x = 14.

    • If you chose B: You added 2 to the 4. The whole bar is 3 parts of 4, which is 12. Then x − 2 = 12. Go to M4.8.1.
    • If you chose C: The whole bar is 3 × 4 = 12, so x − 2 = 12. You took 2 away again, but the 2 must be added back. Go to M4.8.1.
    • If you chose D: The whole bar, 12, is x − 2, not x. Add the 2 back to find x. Go to M4.8.1.

  7. Solve 6(x − 2) = 4(x + 1).

    See the answer

    x = 8

    Expand: 6x − 12 = 4x + 4. Take 4x from both sides: 2x − 12 = 4. Add 12: 2x = 16. Divide by 2: x = 8.

  8. Solve (3x − 2)/4 = 4.

    See the answer

    x = 6

    Multiply both sides by 4: 3x − 2 = 16. Add 2: 3x = 18. Divide by 3: x = 6. Check: (18 − 2)/4 = 4.

Keep it

Copy this into your exercise book.

Open a bracket by multiplying every term inside it. Clear a fraction by multiplying both sides by the number below it. Do the same to both sides.

Draw: Three balances in a row: 3x + 6 against 2x + 10, then x + 6 against 10, then x against 4.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (989 KB). It opens on any phone.