Solving Inequalities

Q: What is the golden rule when solving inequalities?

The golden rule is: when you multiply or divide both sides of an inequality by a negative number, you must flip (reverse) the inequality sign. For example, if −2x > 6, dividing by −2 gives x to <. Forgetting this is the single most common error students make.

Q: How do you show inequalities on a number line?

Use an open circle (○) for strict inequalities ( ) meaning the value is NOT included. Use a filled circle (●) for inclusive inequalities (≤ or ≥) meaning the value IS included. Draw a line or arrow in the direction of the solution set. For combined inequalities like −2 < x ≤ 5, use open at −2 and filled at 5 with a line between.

Q: What is the critical values method for quadratic inequalities?

First solve the quadratic equation (set it equal to zero) to find the critical values. Then sketch a U-shaped parabola through those roots. For > 0 or ≥ 0, the solution is where the curve is above the x-axis (outside the roots). For < 0 or ≤ 0, the solution is where the curve is below the x-axis (between the roots). This visual approach prevents sign errors.

Q: Are quadratic inequalities on the IGCSE Foundation tier?

No. Quadratic inequalities and graphical inequality regions are Higher tier only (grades 7–9). Linear inequalities are tested on both Standard and Higher tiers. Foundation tier students need to solve one-step and two-step linear inequalities and represent solutions on a number line.

Linear, quadratic and graphical inequalities with number line notation, critical values and shading conventions.

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How do you solve inequalities in IGCSE Maths?

Solve linear inequalities using the balance method, just like equations — but flip the inequality sign when you multiply or divide by a negative. Quadratic inequalities (Higher) use the critical values method: solve the equation, sketch the parabola, then read off the required region. Graphical inequalities involve identifying regions on a coordinate plane using solid or dashed boundary lines.

Tier: BothTopic: AlgebraPapers: 1 & 2

Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1

Key Takeaway

The golden rule: multiplying or dividing by a negative number FLIPS the inequality sign. For quadratic inequalities, always sketch the parabola — if the expression is > 0, you need the regions outside the roots; if < 0, you need the region between the roots. On graphs, solid lines mean ≤/≥ and dashed lines mean </>.

How do you solve linear inequalities?

Solving a linear inequality follows the same balance method as solving equations. Add, subtract, multiply or divide both sides to isolate x. The one crucial difference is the golden rule:

Multiply or divide by a negative → FLIP the inequality sign

For example, −3x > 12 becomes x < −4 (dividing by −3 flips > to <).

Number line representation

After solving, you can represent the solution on a number line:

Open circle (○): Used for < or > — the value is not included.
Filled circle (•): Used for ≤ or ≥ — the value is included.

Draw a line or arrow from the circle in the direction of the solution set. For a double inequality such as −2 < x ≤ 5, mark open at −2, filled at 5, and shade between.

How do you solve quadratic inequalities?

Quadratic inequalities are a Higher tier topic. The most reliable approach is the critical values method:

Step 1 — Solve as an equation: Replace the inequality sign with = and solve (usually by factorisation) to find the critical values (roots).
Step 2 — Sketch the curve: Draw a U-shaped parabola passing through the critical values on the x-axis. A positive x² coefficient gives a U shape; negative gives an inverted U.
Step 3 — Identify the region: For > 0 or ≥ 0, the solution is where the curve is above the x-axis (outside the roots). For < 0 or ≤ 0, the solution is where the curve is below the x-axis (between the roots).

Write the final answer using set notation or inequality notation. For "outside the roots" solutions, you need an OR statement: x ≤ a or x ≥ b. For "between the roots", use a combined inequality: a ≤ x ≤ b.

How do you identify graphical inequality regions?

Graphical inequality questions ask you to shade a region on a coordinate grid that satisfies one or more inequalities. Two key rules for the boundary lines:

Solid line: ≤ or ≥ — points on the line are included in the region.
Dashed line: < or > — points on the line are not included.

Quick reference table:

< — less than — open circle on number line — dashed line on graph
≤ — less than or equal to — filled circle — solid line
> — greater than — open circle — dashed line
≥ — greater than or equal to — filled circle — solid line

To decide which side of a boundary line to shade, pick a test point (often the origin (0, 0) if it is not on the line). Substitute into the inequality — if true, shade that side; if false, shade the opposite side. When multiple inequalities are combined, the solution is the region where all shading overlaps.

Worked Examples

Grade 6

Solve 5x − 3 < 2x + 12.

Step 1: Subtract 2x from both sides: 3x − 3 < 12

Step 2: Add 3 to both sides: 3x < 15

Step 3: Divide by 3: x < 5

Answer: x < 5 (open circle at 5, arrow pointing left on number line)

Grade 8/9

Solve x² − 3x − 10 ≥ 0.

Step 1 — Solve as equation: x² − 3x − 10 = 0 ⇒ (x − 5)(x + 2) = 0 ⇒ x = 5 or x = −2

Step 2 — Sketch: Positive x² gives a U-shaped curve crossing the x-axis at −2 and 5.

Step 3 — Identify region: For ≥ 0 we need the curve above or on the x-axis — that is outside the roots.

Answer: x ≤ −2 or x ≥ 5

Grade 8

Shade the region satisfying y ≥ 0, x ≤ 2 and y ≤ x simultaneously.

Step 1: y ≥ 0 means above (or on) the x-axis. Draw a solid line along y = 0.

Step 2: x ≤ 2 means to the left of (or on) the vertical line x = 2. Draw a solid line at x = 2.

Step 3: y ≤ x means below (or on) the line y = x. Draw a solid line y = x through the origin at 45°.

Step 4: The required region is the triangle where all three conditions overlap: above the x-axis, left of x = 2, and below y = x.

Answer: The triangular region with vertices at (0, 0), (2, 0) and (2, 2). All boundary lines are solid.

Common Mistakes

Invalid combined inequality: Writing −2 > x > 5 is mathematically impossible (−2 is not greater than 5). The correct notation for "outside the roots" is x < −2 or x > 5 using two separate statements.

Strict vs inclusive line types: Using a solid line when the inequality is strict (< or >) or a dashed line when it is inclusive (≤ or ≥) loses marks, even if the shading is correct.

Integer solutions: When a question asks to "list the integer values" that satisfy an inequality, you must write them out explicitly. For −3 < x ≤ 2, the integers are −2, −1, 0, 1, 2 (not −3 because the inequality is strict).

Forgetting to flip the sign: Dividing or multiplying by a negative without reversing the inequality is the most common error and loses all accuracy marks.

Exam Tips

Always flip when dividing by a negative: Underline or circle any step where you divide by a negative as a visual reminder to reverse the sign.

Sketch the quadratic curve: For quadratic inequalities, even a rough sketch helps you choose the correct region. Examiners often award a mark for the sketch itself.

Check line types carefully on graphs: Before shading, verify whether each boundary line should be solid or dashed. Mark it clearly — examiners check this first.

Use a test point for graphical regions: Substitute (0, 0) into each inequality. If it satisfies, shade the side containing the origin. This avoids guesswork.

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What is the golden rule when solving inequalities?

The golden rule is: when you multiply or divide both sides of an inequality by a negative number, you must flip (reverse) the inequality sign. For example, if −2x > 6, dividing by −2 gives x < −3. The sign changes from > to <. Forgetting this is the single most common error students make.

How do you show inequalities on a number line?

Use an open circle (○) for strict inequalities (< or >) meaning the value is NOT included. Use a filled circle (●) for inclusive inequalities (≤ or ≥) meaning the value IS included. Draw a line or arrow in the direction of the solution set. For combined inequalities like −2 < x ≤ 5, use open at −2 and filled at 5 with a line between.

What is the critical values method for quadratic inequalities?