Linear inequalities

Inequalities are tested on Papers 1, 2 and 3 of OCR J560. Both solving algebraic inequalities and representing them graphically are assessed. Higher-tier extends to quadratic and simultaneous inequalities.

Inequality notation

Symbol	Meaning
>	greater than
<	less than
≥	greater than or equal to
≤	less than or equal to

Solving linear inequalities

Apply the same operations to both sides as with equations, with one critical exception:

When multiplying or dividing by a NEGATIVE number, FLIP the inequality sign.

Example: −2x > 8 → divide both sides by −2 → x < −4. (Sign flipped!)

Examples:

• 3x + 5 > 11 → 3x > 6 → x > 2.
• 4 − 3x ≤ 10 → −3x ≤ 6 → x ≥ −2 (divided by −3, sign flips).
• 2(x − 3) < x + 5 → 2x − 6 < x + 5 → x < 11.

Number line representation

• Use an open circle (○) for strict inequalities (< or >): the endpoint is NOT included.
• Use a filled/closed circle (●) for non-strict inequalities (≤ or ≥): the endpoint IS included.
• Draw a line/arrow to show all values satisfying the inequality.

Example: x > 2: open circle at 2, arrow pointing right. Example: −3 ≤ x < 5: filled circle at −3, open circle at 5, line between them.

Integer solutions

"List the integers that satisfy 2 < x ≤ 7" → integers greater than 2 and at most 7: 3, 4, 5, 6, 7.

Graphical regions (two variables)

For inequalities in x and y:

1. Draw the boundary line (as y = mx + c) — solid line for ≤ or ≥; dashed for < or >.
2. Test a point (usually (0,0)) to see which side satisfies the inequality.
3. Shade the region that satisfies the inequality (or follow the convention in the question).

Example: y > 2x − 1:

• Draw y = 2x − 1 as a dashed line.
• Test (0,0): 0 > 2(0)−1 = −1 → TRUE → (0,0) is in the required region → shade above the line.

Common OCR exam mistakes

1. Forgetting to flip the sign when multiplying/dividing by a negative number.
2. Using closed circles for strict inequalities on the number line.
3. Shading the wrong region in graphical problems — always test a point.
4. Including the endpoints when listing integers for a strict inequality: 2 < x ≤ 7 does NOT include 2.