Quadratic Inequalities

Grade 7

Why you need the parabola shape

A quadratic inequality like x² − 5x + 6 < 0 asks: for which values of x is the parabola below the x-axis? You cannot solve this purely algebraically without thinking about the graph shape.

The method

Step 1: Solve the quadratic as an equation to find the two roots. Step 2: Sketch the parabola (upside-down ∪ for positive x² coefficient). Step 3: For < 0, the answer is the region between the roots. For > 0, the answer is the region outside the roots.

Remember: positive x² → U-shaped parabola. The parabola is below the x-axis between the roots. That region is where the quadratic is negative.

Worked example

EXAMPLE

Solve x² − 5x + 6 < 0.

Solve x² − 5x + 6 = 0: (x − 2)(x − 3) = 0, so roots are x = 2 and x = 3. U-shaped parabola — below the axis between the roots. 2 < x < 3.

Greater than zero — outside the roots

EXAMPLE

Solve x² − x − 6 > 0.

Roots: (x − 3)(x + 2) = 0, so x = 3 and x = −2. Parabola is above the axis outside the roots. x < −2 or x > 3. Note: this is two separate regions — write it as two inequalities, not one.

Try this one

Solve the inequality

\[ 2x^2 - 5x - 3 \ge 0 \]

[3]

Nova's hint:

This is a quadratic inequality. Factorise the quadratic first to find the critical values where it equals zero.

Ask yourself: 'for an upward parabola with a \( \ge 0 \) sign, do I want the region between the roots or outside them?' A rough sketch of the curve makes the choice clear.

Try to Solve it with Nova?