Solving a Cubic Inequality Graphically

A factorised cubic inequality is solved by tracking the sign of the product across its roots. A quick sketch (or a sign check between roots) shows where the curve is above or below the axis, and the answer is the matching regions.

Solve the inequality $(x + 1)(x - 2)(x - 4) > 0$. [4]

The working

Step 1, find the roots (where each factor is zero): $x = -1, \quad x = 2, \quad x = 4 \quad \text{(M1)}$

Step 2, track the sign across the roots. The cubic has a positive leading coefficient, so it rises to the right (positive for large $x$) and alternates sign at each root. Checking each interval:

(M1, A1)

Step 3, read the regions where the product is $> 0$ (positive): $-1 < x < 2 \quad \text{or} \quad x > 4 \quad \text{(A1)}$

Where the marks are won and lost

• The cubic alternates sign at each simple root. Starting positive on the far right ($x > 4$), it flips at each root moving left.
• The answer is two regions joined by “or”, cubics with three roots give a striped pattern, so "$> 0$" picks out the positive stripes.
• A quick sketch or the sign table prevents guessing; testing a point in each interval confirms the signs.

Common mistakes

• Giving only one region instead of both positive stripes.
• Getting the sign pattern backwards (check the far-right interval first, it’s positive for a positive cubic).
• Including the roots (use strict inequalities for "$>$", roots excluded).

Full method: Solving Cubic Inequalities Graphically notes. Topic home: Equations, Inequalities and Graphs pillar.