Solving Cubic Inequalities Graphically

A cubic inequality like $(x + 2)(x - 1)(x - 3) > 0$ cannot be safely solved by sign-rule memory, three factors generate too many cases. The graph does it in seconds, and 0606 expects exactly that method.

The method

Solve $(x + 2)(x - 1)(x - 3) > 0$.

1. Roots: $x = -2, 1, 3$ (from the factors, or run the cubic factorising routine first if given expanded form).
2. Sketch: positive $x^3$ coefficient, comes from below-left, ends up-right, weaving through the three crossings.
3. Read "$> 0$": the curve is above the axis between $-2$ and $1$, and after $3$. $-2 < x < 1$ or $x > 3$

The sketch needs thirty seconds and no scale, just the crossings in order and the end behaviour. Negative $x^3$ coefficient flips the weave (starts up-left, ends down-right); check the sign before drawing, because every region answer depends on it.

Repeated roots change the reading

A squared factor touches the axis instead of crossing:

$(x - 2)^2(x + 1) > 0$: roots at $-1$ and $2$ (double). The curve crosses at $-1$, then touches at $2$ without going below. Answer: $x > -1$, $x \ne 2$, the touch-point itself gives equality, excluded by the strict $>$.

That "$x \ne 2$" exclusion is a genuinely tested detail, and only the sketch makes it visible. With $\ge$, the answer becomes simply $x \ge -1$.

Writing the regions

Same conventions as quadratic inequalities: separate intervals joined by “or”; sandwich notation for bounded regions; strictness following the question’s symbol (strict inequality excludes the roots, non-strict includes them).

Exam connections

These inequalities almost always arrive as the second part of a question whose first part factorised the cubic, a hence chain. Re-deriving the roots from scratch wastes the gift. The same sketch also answers “for what values of $k$ does $p(x) = k$ have three solutions?”, slide a horizontal line between the local max and min values.

Common mistakes

• Regions guessed from sign rules instead of drawn
• End behaviour assumed positive without checking the $x^3$ coefficient
• Repeated roots drawn as crossings
• Strict inequalities including the roots (or the touch-point)
• Three-root cubics answered with a single interval, almost always wrong

Full topic context: Equations, Inequalities & Graphs notes.