Inequalities and number lines
Maths · Key Stage 3 · The School
An inequality answers with a range
An equation like x + 3 = 7 has one answer. An inequality like x + 3 > 7 has infinitely many: every number above 4. So the answer is not a value but a description of a set, written x > 4, and that shift is the whole idea of the topic.
The circle tells you whether the end is included
On a number line, an open circle at 4 means 4 itself is excluded (x > 4); a filled circle means it is included (x ≥ 4). The arrow shows which way the set runs. Getting the circle wrong changes the answer by exactly one number, and that number is the one examiners test.
Solve it like an equation, with one exception
Add, subtract, multiply and divide both sides exactly as with an equation. 2x + 1 ≤ 9 gives 2x ≤ 8 gives x ≤ 4. Everything transfers across from equations unchanged, which makes the single exception below easy to forget and easy to test.
Multiplying or dividing by a negative flips the sign
Start from something true: 3 < 5. Multiply both sides by −1 and you get −3 and −5. But −3 is GREATER than −5, so the statement only stays true if the sign turns round: −3 > −5. So when you divide −2x < 6 by −2, you must write x > −3, not x < −3. Check by substituting: x = 0 gives −0 < 6, true, and 0 > −3, consistent.
Two ends at once, and counting the whole numbers inside
Written −1 < x ≤ 3, that is two inequalities in one: x is above −1 and at or below 3. On the line it is an open circle at −1, a filled circle at 3, and the region between them shaded. Questions then often ask for the INTEGER values that satisfy it, which is a different answer again: 0, 1, 2 and 3. Not −1, because that circle is open. Four values — and the one most often lost is 3, the boundary people shade correctly and then forget to count.