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Step 9: Proportional reasoning

Two-step equations and inequalities

Step 9 · Proportional reasoning topics

Solve equations such as 3x + 5 = 20 by undoing steps in reverse order, solve inequalities (turning the sign when dividing by a negative), and write both from word problems.

Solving two-step equations

In 3x + 5 = 20, two things were done to x: it was multiplied by 3, then 5 was added. Undo them in reverse order: first take away 5, then divide by 3. Whatever you do, do it to both sides.

3x + 5 = 20: both pans weigh the same. Take 5 from each pan, then share what is left into 3.

Example: Solving

Solve 3x + 5 = 20.

  1. Subtract 5 from both sides: 3x = 15.
  2. Divide both sides by 3: x = 5.
  3. Check: 3 × 5 + 5 = 20.

Answerx = 5

Example: With a negative

Solve −2x + 7 = 1.

  1. Subtract 7: −2x = −6.
  2. Divide by −2: x = 3.

Answerx = 3

Example: With parentheses

Solve 4(x − 2) = 28.

  1. Divide both sides by 4: x − 2 = 7.
  2. Add 2: x = 9.

Answerx = 9

Inequalities

An inequality says one side is bigger. Solve it like an equation, with one change: when you multiply or divide both sides by a negative number, turn the sign round.

Example: An inequality

Solve 2x − 3 > 7.

  1. Add 3: 2x > 10.
  2. Divide by 2: x > 5.

Answerx > 5

x > 5: an open dot at 5, because 5 itself does not work, and everything to its right.

Example: Turning the sign

Solve −3x ≤ 12.

  1. Divide both sides by −3 and turn the sign: x ≥ −4.

Answerx ≥ −4

Writing equations from words

To turn words into an equation, name the unknown with a letter, then write each fact as maths.

Example: From words

A gym charges $20 to join and $15 a month. Jo has paid $110 in all. How many months has she paid for?

  1. Let m be the months: 20 + 15m = 110.
  2. Subtract 20: 15m = 90.
  3. Divide by 15: m = 6.

Answer6 months

Example: An inequality from words

Tickets cost $12 each and you have $50. How many tickets can you buy?

  1. 12t ≤ 50.
  2. t ≤ 50 ÷ 12, which is about 4.17.
  3. Only whole tickets count, so at most 4.

Answer4 tickets

Try it yourself

Work each one on paper first, then open the answer to check.

  1. Solve 5x − 4 = 21.

    Show answer

    x = 5

  2. Solve x3+2=9\frac{x}{3} + 2 = 9.

    Show answer

    x = 21

  3. Solve −4x + 1 = 13.

    Show answer

    x = −3

  4. Solve 3x + 4 ≥ 19.

    Show answer

    x ≥ 5

  5. Solve −2x < 8.

    Show answer

    x > −4 (the sign turns because you divide by −2)

  6. A taxi charges $3 plus $2 a mile. A ride cost $19. How many miles was it?

    Show answer

    8 miles: 3 + 2m = 19, so 2m = 16.

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