Maths › Further Pure 1 › Inequalities and inequations
Inequalities and inequations
Never multiply an inequality by something whose sign you do not know. Move everything to one side, factorise, and let a sign diagram or a sketch read off the answer.
Builds on Simultaneous equations and inequalities and Functions, inverses and the modulus.
IN THIS TOPIC
- Solve rational inequalities by collecting on one side and using a sign diagram.
- Handle modulus inequalities by squaring or by splitting into cases.
- State solution sets correctly, excluding the values that break a denominator.
WHAT YOU PROBABLY THINK
To solve x/(x − 2) < 3 you simply multiply both sides by x − 2 and solve the linear inequality.
Why the obvious move is fatal
Multiplying an inequality by a negative number reverses it, and x − 2 is negative for x < 2. The shortcut in the opening claim silently assumes a sign it has no right to, and loses the entire interval x < 2. The safe route: subtract, combine over a common denominator, and factorise.
WORKED EXAMPLE
The right way round
Solve x/(x − 2) < 3.
Subtract: x/(x − 2) − 3 < 0, so (x − 3(x − 2))/(x − 2) < 0, that is (6 − 2x)/(x − 2) < 0.
The critical values are x = 2 and x = 3. Testing each interval: negative for x < 2, positive between, negative for x > 3.
Solution: x < 2 or x > 3. The naive multiplication would have returned only x > 3, throwing away half the answer.
Critical values come from zeros of the numerator and of the denominator alike, but they behave differently: a numerator zero may be included when the inequality is weak, while a denominator zero is always excluded, since the expression has no value there.
Modulus and the sketching route
For an inequality between a modulus and something non-negative, squaring is legitimate and often quickest, since both sides are then non-negative and the direction is preserved. Otherwise, split into cases on the sign inside the modulus, or sketch both sides and read the intervals where one curve sits above the other.
WORKED EXAMPLE
A modulus against a line
Solve |x² − 1| > 2(x + 1).
Where x² ≥ 1 the modulus opens directly: x² − 1 > 2x + 2 gives x² − 2x − 3 > 0, that is (x − 3)(x + 1) > 0, so x > 3 (within this case) or x < −1.
Where x² < 1 it opens with a sign change: 1 − x² > 2x + 2 gives x² + 2x + 1 < 0, which is (x + 1)² < 0: impossible.
Solution: x < −1 or x > 3. The sketch agrees: the V-shaped modulus graph rises above the line exactly outside those two crossings.
TRY IT UNSEEN
Two fractions at once
Solve 1/(x − 1) > x/(x + 2).
Show the working
Subtract and combine: (x + 2 − x(x − 1))/((x − 1)(x + 2)) > 0, which is (2 + 2x − x²)/((x − 1)(x + 2)) > 0.
The numerator vanishes at x = 1 ± √3 and the denominator at x = 1 and x = −2, giving four critical values.
Testing the five intervals: the solution is −2 < x < 1 − √3 or 1 < x < 1 + √3. Both denominator zeros stay excluded throughout.
THE EXAM BIT
- Never multiply by an expression of unknown sign; subtract to one side instead.
- List every critical value, from numerator and denominator, before testing intervals.
- Exclude denominator zeros from the solution set even when the inequality is weak.
- A quick sketch of both sides is worth the thirty seconds; it catches lost intervals immediately.
CHECK YOURSELF
Solve (x + 1)/(x − 3) ≥ 0.
Show a hint
Critical values −1 and 3; test the three intervals and mind which endpoint can be included.
Show the answer
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Collect on one side, factorise, and read a sign diagram; multiplying by an unknown sign loses intervals.
Modulus inequalities: square when both sides are non-negative, or split into cases and sketch to check.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Solve rational inequalities by collecting on one side and using a sign diagram.
- Handle modulus inequalities by squaring or by splitting into cases.
- State solution sets correctly, excluding the values that break a denominator.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.