Algebra · step 5 of 13

Inequalities & absolute value

Inequalities behave like equations with two crucial exceptions: multiplying or dividing by a negative reverses the sign, and you must never multiply by a variable expression whose sign is unknown. Absolute values split into cases — |x| < a means −a < x < a, while |x| > a means x < −a or x > a — and quadratic inequalities are solved by finding the roots and testing intervals.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Flip on multiplication by a negativeDividing −2x > 6 by −2 gives x < −3. This is the single most common lost mark.
Never cross-multiply by a variableFor 1/x > 2, move everything to one side and analyse sign instead — x could be negative.
Split absolute values into cases|expression| < a becomes a double inequality; |expression| > a becomes a union of two.
Solve quadratics by root and sign testFind where the quadratic is zero, then test a point in each resulting interval and read off where the sign matches.

Worked example

Solve x² − x − 6 > 0.

Factor: (x − 3)(x + 2) > 0, with roots at x = 3 and x = −2.
These split the line into (−∞,−2), (−2,3) and (3,∞).
Test x = −3: (−6)(−1) = 6 > 0 ✓. Test x = 0: (−3)(2) = −6 < 0 ✗.
Test x = 4: (1)(6) = 6 > 0 ✓.

Answer. x < −2 or x > 3, i.e. (−∞, −2) ∪ (3, ∞).

Where marks get dropped

These are the specific errors that cost credit on inequalities & absolute value questions — QED's rubric penalises each of them separately.

Forgetting to reverse the inequality when multiplying or dividing by a negative number.
Multiplying an inequality by x without knowing its sign, which silently assumes x > 0 and loses half the solution set.
Writing a quadratic solution as a single interval when the answer is a union. For > 0 with two real roots, the solution is the OUTSIDE of the roots.

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Inequalities & absolute value — frequently asked questions

How do I solve |2x − 1| < 5?

Rewrite as −5 < 2x − 1 < 5, add 1 throughout to get −4 < 2x < 6, then divide by 2: −2 < x < 3.

Why can’t I cross-multiply 1/x > 2?

Because multiplying by x flips the inequality when x < 0. Rearrange to (1 − 2x)/x > 0 and do a sign analysis instead — the answer is 0 < x < 1/2.

How should I write the answer?

Interval notation or an inequality, consistently. Use ∪ for unions and be careful whether endpoints are included: strict inequalities exclude them.

The rest of Algebra

Foundational algebra to close prerequisite gaps. Each subtopic below has its own method, worked example and mark-losing traps.

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