Quadratic Inequalities
Grade 7Why 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
\[ 5x - 4 \geq 11 \]
[2]
Nova's hint:
Try to Solve it with Nova?
This is a linear inequality — solve it the same way you would a linear equation.
Ask yourself: 'what inverse operations do I need to isolate \(x\)?'
For example, for \( 4x - 3 \geq 9 \), add 3 to both sides to get \( 4x \geq 12 \), then divide by 4.
