2.5 Solving Quadratic Inequalities

2026 📋 Syllabus Objectives

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

  • Find the solution set for quadratic inequalities using a graphical method
  • Find the solution set for quadratic inequalities using an algebraic method
  • Write your answers in the correct notation, for example:
    • −3 < x < 4 (a single connected range)
    • x < 1 or x > 6 (two separate ranges)

📖 Main Content

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What is a Quadratic Inequality?

You already know how to solve a linear inequality — an inequality where the highest power of x is 1. For example:

Solve 2(x − 5) < 9 → 2x − 10 < 9 → 2x < 19 → x < 9.5

Solve 5 − 3x ≥ 17 → −3x ≥ 12 → x ≤ −4 (when you divide or multiply by a negative number, the inequality sign flips)

A quadratic inequality works the same way in principle, but the highest power of x is 2. This means you are looking at a curved graph (a parabola) rather than a straight line, and the solution is often a range of values rather than just one boundary.

Examples of quadratic inequalities:

  • x² − x − 6 < 0
  • x² − 5x + 4 ≥ 0
  • 2x² + 3x − 2 > 0

Key Idea: The Parabola and the x-axis

When you solve a quadratic inequality, you are essentially asking:

"For which values of x is the quadratic expression above, below, or on the x-axis?"

  • If the inequality is < 0 or ≤ 0, you want the part of the curve that is below (or on) the x-axis.
  • If the inequality is > 0 or ≥ 0, you want the part of the curve that is above (or on) the x-axis.

The roots (also called the critical values) are where the quadratic equals zero — these are the boundary points of your solution.

Method 1: Graphical Method

Step 1: Set the quadratic equal to zero and find the roots (the x-intercepts of the graph).

Step 2: Sketch the parabola — note whether it opens upward (positive x² coefficient) or downward (negative x² coefficient).

Step 3: Look at the sketch to decide which region of x satisfies the inequality.

🔍 Worked Example 1 (Graphical) — "Less Than" Inequality

Solve x² − x − 6 < 0

Step 1 — Find the roots:

Set x² − x − 6 = 0

Factorise: (x − 3)(x + 2) = 0

So x = 3 or x = −2

Step 2 — Sketch the parabola:

The coefficient of x² is +1 (positive), so the parabola opens upward (U-shape).

The curve crosses the x-axis at x = −2 and x = 3.

         y
         |
         |     /         \
         |    /           \
  ───────|──/─────────────\──── x
        -2                 3
         |  (below x-axis) |
         |_________________|

Step 3 — Read the solution:

We want where the curve is below the x-axis (< 0).

Looking at the sketch, the curve dips below the x-axis between x = −2 and x = 3.

✅ Solution: −2 < x < 3

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