Key ideas
- Keep the equation balanced: whatever you do to one side, do to the other.
- Brackets? Multiply out first. Watch the signs: −2(x − 3) = −2x + 6.
- Fractions? Multiply every term on both sides by the lowest common denominator to clear them.
- Collect the x terms on one side and the numbers on the other, then divide.
- Answers that don't work out exactly stay as a fraction in its simplest form, like x = 177. A rounded decimal is not exact.
- An inequation uses <, >, ≤ or ≥ instead of =. Solve it in exactly the same way, with one rule:
- If you multiply or divide by a negative number, reverse the inequality sign. Or avoid this by collecting the x terms on the side where they stay positive.
- Exam questions on this topic are usually 3 marks: •1 start to process (expand brackets or clear fractions), •2 rearrange to ax = b (or ax > b), •3 solve. A correct answer with no working gets no marks.
Open circle: 3 is not included.
Filled circle: −1 is included.
Unknown on both sides
Solve the equation 5x + 3 = 2x + 18.(2 marks)
Collect the x terms.5x − 2x + 3 = 183x + 3 = 183x = 15✓1
In plain EnglishTake 2x from both sides so the x terms are all on the left. Then take 3 from both sides so the numbers are on the right.
Divide.x = 153x = 5✓2
In plain EnglishDivide both sides by 3.
Check: 5 × 5 + 3 = 28 and 2 × 5 + 18 = 28. Both sides match, so x = 5 is right.
Brackets
Solve the equation 7x − 2 = 3(x + 4).(3 marks)
Expand the bracket.7x − 2 = 3x + 12✓1
In plain EnglishMultiply both terms inside the bracket by 3. This is the first mark: starting to process.
Rearrange.7x − 3x = 12 + 24x = 14✓2
In plain Englishx terms to the left, numbers to the right. Moving a term across the equals sign changes its sign.
Solve.x = 144x = 72✓3
In plain EnglishDivide by 4, then simplify the fraction. x = 3.5 is fine too, because it is exact.
Fractions
Solve the equation 2x − 13 − x4 = 2.(3 marks)
Clear the fractions.
The denominators are 3 and 4, so multiply every term by 12.
12 × 2x − 13 − 12 × x4 = 12 × 24(2x − 1) − 3x = 24✓1In plain English12 ÷ 3 = 4 and 12 ÷ 4 = 3. Every term gets multiplied, including the 2 on the right. Keep the bracket so the whole numerator is multiplied.
Expand and rearrange.8x − 4 − 3x = 245x = 28✓2
In plain EnglishMultiply out, collect the x terms, and move the number across.
Solve.x = 285✓3
In plain EnglishDivide by 5. 28/5 doesn't simplify, so leave it as a fraction (or 5.6, which is exact).
⚠ Multiply every term. The most common mistake is forgetting to multiply the number on the right (here the 2) by 12.
Inequations
Solve, algebraically, the inequation 3(x − 2) + 5 > x + 9.(3 marks)
Expand the bracket.3x − 6 + 5 > x + 9✓1
In plain EnglishWork exactly as you would with an equation, keeping the inequality sign in place of the equals sign.
Rearrange.3x − x > 9 + 6 − 52x > 10✓2
In plain EnglishCollect x terms on the left and numbers on the right.
Solve.x > 102x > 5✓3
In plain EnglishDivide by 2. It's a positive number, so the sign stays the same.
Negative x terms
Solve, algebraically, the inequation 11 − 2(x + 3) < 3x − 10.(3 marks)
Expand the bracket.11 − 2x − 6 < 3x − 10✓15 − 2x < 3x − 10
In plain EnglishThe −2 multiplies both terms in the bracket: −2 × x = −2x and −2 × 3 = −6.
Rearrange so the x term stays positive.5 + 10 < 3x + 2x15 < 5x✓2
In plain EnglishHere the x terms are collected on the right, where they are positive. Then nothing needs reversing.
Solve.3 < xx > 3✓3
In plain English3 is less than x means the same as x is greater than 3. Writing it with x first is clearer.
The other way: collect on the left to get −5x < −15. Now you divide by −5, a negative number, so the sign reverses: x > 3. Same answer. Both methods get full marks.
Inequations with fractions
Solve, algebraically, the inequation x + 32 − 2 > 3x5.(3 marks)
Clear the fractions.
Multiply every term by 10.
10 × x + 32 − 10 × 2 > 10 × 3x55(x + 3) − 20 > 6x✓1In plain English10 is the lowest common multiple of 2 and 5. Don't forget the −2: 10 × 2 = 20.
Rearrange.5x + 15 − 20 > 6x−5 > 6x − 5x−5 > x✓2
In plain EnglishCollect the x terms on the right, where there are more of them, so the x term stays positive.
Solve.x < −5✓3
In plain English−5 is greater than x means x is less than −5. When you swap the sides round, the sign turns round too.
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Your turn
Non-calculator. Type answers like x = 5, x = 17/7 or x > -3 (use < > <= >=). Leave answers that don't work out exactly as fractions. Stuck? Tap Need a hint?
Past paper questions
Ready for exam-style questions? Every National 5 question on this topic so far is worth 3 marks.
National 5 linear equationsEvery National 5 equations question in one PDF (2016 to 2025), with answers. Several include fractions.maths.scot · PDF Open ↗ (opens in a new tab) National 5 linear inequationsEvery National 5 inequations question in one PDF (2015 to 2026), with answers.maths.scot · PDF Open ↗ (opens in a new tab) Equations: topic pageEach question separately, with worked videos.maths.scot Open ↗ (opens in a new tab) Inequations: topic pageEach question separately, with worked videos.maths.scot Open ↗ (opens in a new tab)