Inequalities Topic Review summary

Key learning points

• Bring to one side; find zeros and discontinuities as critical points.
• Sign chart / test intervals to solve $f(x)>0$ etc.
• For rational inequalities, never multiply by a quantity of unknown sign blindly — consider cases or a common-denominator function.
• Include or exclude endpoints per $<$ vs $\\le$.
• Express answers in interval notation or inequalities as required.

Worked example 1 — Quadratic

$x^2-x-6>0$.

Working

Roots $-2,3$; parabola up.

Worked example 2 — Rational

$\\frac{x-1}{x+2}\\ge0$.

Working

$x=-2$ undefined.

Worked example 3 — Practice point 1

Apply this idea from the Topic Review: Bring to one side; find zeros and discontinuities as critical points.

Working

Attempt without notes, then check the answers PDF.

Worked example 4 — Practice point 2

Sign chart / test intervals to solve $f(x)>0$ etc.

Working

Attempt without notes, then check the answers PDF.

Scaffolded Learning Sheet and answers (PDFs)

This page is a short, searchable overview for revision and search engines. Use the Scaffolded Learning Sheet first, then the Answers PDF for a full written solution. The printable PDFs remain the main classroom resource.