4.5 Solving Cubic Inequalities

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Solve cubic inequalities graphically of the form f(x) ≥ d, f(x) > d, f(x) ≤ d, and f(x) < d, where f(x) is a product of three linear factors and d is a constant.

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What is a Cubic Inequality?

A cubic inequality is a mathematical statement that compares a cubic expression (an expression where the highest power of x is 3) to a value, using inequality signs like ≥, >, ≤, or <.

The type of cubic inequality you need to solve looks like this:

f(x) ≥ d, f(x) > d, f(x) ≤ d, or f(x) < d

Where:

  • f(x) is a cubic expression that is a product of three linear factors — meaning it is written as three brackets multiplied together, like (x − a)(x − b)(x − c), possibly with a constant k in front.
  • d is just a plain number (a constant), like 0, 1, 2, −3, etc.

Examples of what f(x) looks like:

  • (x − 1)(x − 3)(x + 2)
  • x(x − 5)(x + 1)
  • ½(x − 2)(x − 4)(x + 1)

Why Use a Graphical Method?

Cubic inequalities are much harder to solve using pure algebra compared to linear or quadratic inequalities. The graphical method makes things much more visual and manageable.

The idea is simple:

  • You draw (or are given) the graph of the cubic curve y = f(x).
  • You draw the horizontal line y = d.
  • You then look at where the curve is above, below, or on that line — and that tells you your answer.

Understanding the Graph of a Cubic Function

Before solving inequalities, you need to understand what the graph looks like.

A cubic function written as f(x) = k(x − a)(x − b)(x − c) crosses the x-axis at three points:

  • x = a
  • x = b
  • x = c

These are called the roots (or x-intercepts) — the points where the curve touches or crosses the x-axis (where y = 0).

To find the y-intercept (where the curve crosses the y-axis), substitute x = 0 into the equation.

The shape of the curve depends on whether k is positive or negative:

  • If k > 0 (positive): the curve goes from bottom-left to top-right (like a stretched S-shape going upward).
  • If k < 0 (negative): the curve goes from top-left to bottom-right (like an upside-down S-shape).

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