Solving Linear Inequalities

SPEC A22 • ALGEBRA

⭐ High-priority. Foundation (Additional) + Higher. Solve, represent on a number line, and — at Higher — use set notation, handle quadratic inequalities, and shade regions on a graph.

The One Extra Rule

Solving a linear inequality is almost identical to solving a linear equation. All the same operations apply — with one crucial extra rule: when you multiply or divide by a negative, you must flip the inequality sign.

Authentic exam evidence

Jun 2023 P1H Q2: "Solve 2x < 26" — 1 mark. Candidates who gave x = 13 (no inequality symbol) lost the mark.

Nov 2024 P1H Q15: "Solve 20 − 5x ≤ 30" — 3 marks. Most students earned at least one mark, but over half lost marks on the sign-flip step when dividing by a negative.

Nov 2022 P1H Q25: "Solve x² − 5x − 6 < 0" — Higher-only quadratic inequality, 2 marks.

Tier note

Foundation (Additional): solving in one variable, number-line representation. Higher only: set notation, quadratic inequalities, and two-variable graphical regions.

Key Terms & Symbols

SymbolMeaningStrict or inclusive?

<less thanStrict (value not included)

>greater thanStrict (value not included)

≤less than or equal toInclusive (value is included)

≥greater than or equal toInclusive (value is included)

Solution set

The set of all values that satisfy the inequality. Unlike an equation (which typically has one answer), an inequality usually has infinitely many answers — shown as a range or on a number line.

Memory hook for ≤ and ≥

The extra line underneath the symbol is like a "fill-in line" — it tells you to fill in (close) the circle on a number line. Strict inequalities (<, >) have no underline → open circle.

Same Rules — One Exception

Everything you learned for solving equations applies to inequalities:

✓ Same as equations

Add or subtract the same thing to/from both sides.

Multiply or divide both sides by a positive number.

Expand brackets, collect like terms, clear fractions.

The one big difference

Multiply or divide by a negative → FLIP the inequality sign.

Example: −2 < 4 is true. Multiply both sides by −1 without flipping: 2 < −4, which is false. To keep it true, flip the sign: 2 > −4 ✓.

Basic Solving — Positive Coefficient

When the coefficient of x stays positive throughout, inequalities work exactly like equations — just keep the sign as-is.

Example 1 — Jun23 P1H Q2 authentic

2x < 26 (÷ 2)

x < 13

Example 2 — Jun24 P1H Q10(b) authentic

5y + 14 ≥ 11 (− 14)

5y ≥ −3 (÷ 5)

y ≥ −3/5

The final answer form

The mark scheme explicitly requires the inequality symbol in the final answer — not just a number. Writing "x = 13" when the answer is x < 13 scores zero even if the arithmetic is right (Jun 2023 P1H Q2 MS: "x < 13 in working with x = 13 on answer line: B0").

The Sign-Flip Rule

Whenever you multiply or divide by a negative number, the inequality sign flips direction: < becomes >, ≤ becomes ≥, and vice versa.

Why does this happen?

On a number line, 2 is greater than 1. But −2 is less than −1. Multiplying by a negative reverses the order of numbers — so any inequality that compared them must also reverse.

Alternative — avoid the flip entirely

You can often avoid the flip by adding x -terms to both sides to make the coefficient positive. Example: −3x > 12 → add 3x to both sides → 0 > 12 + 3x → subtract 12 → −12 > 3x → x < −4. Same answer, no flip needed. Whichever feels safer — pick one and stick with it.

Number Line Conventions

The spec is explicit: open circle (hollow) for strict inequalities, closed circle (filled) for inclusive inequalities. Arrow shows which direction the solution extends.

All four variants at the boundary x = 3. Open circle excludes the value (strict); closed circle includes it (inclusive). Arrow direction shows the solution range.

Compound Inequalities

A compound inequality combines two conditions into one statement: e.g., −2 ≤ x < 5 means x is at least −2 AND strictly less than 5.

Closed circle at −2 (included), open circle at 5 (excluded), line connecting.

Solving a compound — do every step throughout

Solve: −2 ≤ 3x + 1 < 10

(subtract 1 throughout)

−3 ≤ 3x < 9

(divide by 3 throughout)

−1 ≤ x < 3

MCE Chain for Inequalities

Every inequality follows this three-stage process. Mark schemes award marks at each stage.

MMethod. Treat it like an equation: expand brackets, collect variable terms, isolate x. Watch for any step that multiplies or divides by a negative — those need a sign flip.

CCalculate. Apply inverse operations, one per line. When you flip, mark it clearly: write "(÷ −2, flip)" next to that step.

EEvaluate. Final answer must include the inequality symbol (e.g., x ≥ −2, not just −2). If asked, represent on a number line with the correct circle type and arrow direction.

Worked Example: Setup

Ex

Higher — 3 marks (Nov 2024 P1H Q15(a) authentic)

Solve the inequality 20 − 5x ≤ 30.

M — Method

The coefficient of x is −5 (negative). Two valid approaches:

Route A (flip): Subtract 20, leaving −5x ≤ 10. Divide by −5 and flip the inequality.

Route B (avoid the flip): Add 5x to both sides first, making the coefficient positive. No flip needed — same answer.

Worked Example: Calculate & Evaluate

C — Calculate (Route A, with flip)

20 − 5x ≤ 30

(subtract 20)

−5x ≤ 10

(÷ −5, FLIP)

x ≥ −2

E — Evaluate (check)

Test x = 0 (should satisfy):

20 − 5(0) = 20, and 20 ≤ 30 ✓

Test x = −3 (should not):

20 − 5(−3) = 35, and 35 ≤ 30 is false ✓ (correctly excluded)

Final answer: x ≥ −2

Mark breakdown (Nov 2024 mark scheme)

M1 for rearrangement with −5x the subject (or 5x positive). M1dep for correct rearrangement with x positive (dependent on first M1). A1 for x ≥ −2 (or equivalently −2 ≤ x). SC1 (special case) for just "−2" without the inequality symbol — so partial credit is possible, but the inequality is essential for full marks.

Set Notation Higher

At Higher, you may be asked to express the solution using set notation. Two common forms appear in AQA resources:

Set-builder notation

{ x : x > 3} — reads "the set of all x such that x is greater than 3". The colon (or sometimes a vertical bar) means "such that".

Interval notation

InequalityIntervalMeaningx > 3(3, ∞)Round bracket = strict (excluded)x ≥ 3[3, ∞)Square bracket = inclusive (included)−2 ≤ x < 5[−2, 5)Mixed brackets match inequality type

What the exam expects

AQA questions typically give you set notation in the question stem and ask you to interpret or produce solutions using the same form. The spec requires you to recognise and use set notation at Higher.

Quadratic Inequalities Higher

A quadratic inequality like x² − 5x − 6 < 0 asks: "for which x is the parabola below the x-axis?"

1Factorise to find the roots: x² − 5x − 6 = (x + 1)(x − 6) = 0 → x = −1 or x = 6.

2Sketch the parabola. Coefficient of x² is positive → opens upward → dips below the x-axis between the roots.

3For x² − 5x − 6 < 0 we want where the parabola is BELOW the axis: solution is −1 < x < 6.

The parabola y = x² − 5x − 6 cuts the x-axis at x = −1 and x = 6. Between these roots the curve dips below the x-axis — that's where y < 0.

Two-Variable Graphical Regions Higher

At Higher, you may be asked to identify or draw a region satisfying one or more inequalities on a coordinate grid (e.g. Jun 2022 P1H Q23 — "identify the region satisfying x > 3 AND y > 1 AND x + y ≤ 7").

Line conventions (spec-required)

Strict inequality (< or >):dashed line — the line itself is NOT included.

Inclusive inequality (≤ or ≥):solid line — the line IS included.

Region R: y > 2x + 1. Dashed line (strict); shaded region above. Test point (0, 5): 5 > 2(0) + 1 = 1 ✓ — in the region.

Test a point to check which side

If unsure which side to shade, substitute a point NOT on the line (the origin (0, 0) is often easiest). If the point satisfies the inequality, shade its side. If not, shade the other side.

Weak vs Strong: The Sign Flip

When Nov 2024 P1H Q15 asked "Solve 20 − 5x ≤ 30", many candidates lost marks on the flip step. This is the single biggest inequality trap at GCSE.

Weak student

Solves −5x ≤ 10 as x ≤ −2. Divided by −5 correctly but kept the sign the same.

Lands on the wrong solution set — loses the A1.

Strong student

Solves −5x ≤ 10 as x ≥ −2. Recognised the division by −5 and flipped the inequality sign.

Gets the correct solution set. Full marks.

Quick self-check

After solving, pick a value that should satisfy your answer and substitute. For x ≥ −2, try x = 0: is 20 − 5(0) = 20 ≤ 30? Yes ✓. Try x = −3 (which should fail): 20 − 5(−3) = 35, and 35 ≤ 30 is false ✓ (correctly excluded). Confidence built.

Examiner Alerts

Nov 2024 P1H Q15 — the flip

Over half of candidates lost marks when solving 20 − 5x ≤ 30 — they divided by −5 but forgot to flip. Whenever you divide (or multiply) by a negative, FLIP. Mark it clearly in your working: "(÷ −5, flip)".

Jun 2023 P1H Q2 — keep the symbol

On "Solve 2x < 26" (1 mark), candidates who wrote x = 13 instead of x < 13 scored zero. The mark scheme explicitly states: "x = 13 in working with x < 13 on answer line: B0". The final answer MUST include the inequality symbol.

Wrong circle on number line

Filled circle for strict (or open for inclusive) is the second-most common number-line error. Remember: ≤ and ≥ have a line underneath — think of that line as "filling in" the circle.

Forgot the arrow

A number-line representation needs a visible arrow showing the direction of the solution set — not just a circle. A circle with no arrow rarely scores.

Practice Question 1

Basic solve (Jun 2023 P1H Q2 authentic). Solve 2x < 26. [1 mark]

M

Divide both sides by 2 (positive, so no flip).

C

2x < 26 → x < 13.

E

Final answer: x < 13. Mark scheme: "x < 13 or 13 > x B1". Writing just "13" or "x = 13" scores zero.

Practice Question 2

Flip practice. Solve 4 − 2x > 12 and represent the solution on a number line. [3 marks]

M

Subtract 4 from both sides, then divide by −2 (FLIP the sign).

C

4 − 2x > 12 → −2x > 8 → x < −4 (÷ −2, flipped > to <).

E

Check x = −5: 4 − 2(−5) = 14 > 12 ✓. Number line: open circle at −4, arrow pointing left (strict inequality, solution extends to negative infinity).

Practice Question 3 Higher

Quadratic inequality (Nov 2022 P1H Q25 authentic).

(a) Solve x² − 5x − 6 < 0. [2 marks]

(b) Show your solution on a number line. [1 mark]

(a) M

Factorise: x² − 5x − 6 = (x + 1)(x − 6). Set equal to 0 to find the boundary points.

(a) C

Roots: x = −1 or x = 6. The parabola opens upward (positive x² coefficient), so y < 0 between the roots.

(a) E

Solution: −1 < x < 6. Mark structure: M1 for correct factorisation or finding both roots; A1 for the correct solution range.

(b)

Number line: open circles at −1 and 6 (strict inequality), line connecting them. B1 for the correct number line. Both ends are open because the original inequality is strict (<, not ≤).

Summary

The One Extra Rule

Multiply or divide by a negative → FLIP the inequality sign.

Number Line Conventions

Open circle: strict (< or >) — value excluded

Closed (filled) circle: inclusive (≤ or ≥) — value included

Arrow direction: shows the range of the solution set

Foundation (Additional)

Solve one-variable inequalities

Represent solution on number line

Compound inequalities (both ends)

Higher extensions

Set notation & interval notation

Quadratic inequalities (use parabola sketch)

Two-variable graphical regions (dashed vs solid lines)

Top Exam Tip

The #1 inequality mistake at GCSE is missing the sign flip when dividing by a negative. If your working ever involves dividing or multiplying by a negative, write "(flip)" next to that step — a tiny habit that secures a lot of marks.