First solve the equation to find the critical values. x² + 8x + 15 = (x + 3)(x + 5), so the roots are −5 and −3.
Sketch the parabola. It is U-shaped, so it is below the x-axis (≤ 0) between the roots and above it (> 0) outside them. So x² + 8x + 15 ≤ 0 gives −5 ≤ x ≤ −3, while x² + 8x + 15 > 0 gives x < −5 or x > −3.
Worked example
Solve x² + 3x − 4 ≤ 0
- Factorise: (x + 4)(x − 1), so the roots are −4 and 1.
- A U-shaped curve is below the axis between the roots.
Answer:−4 ≤ x ≤ 1
Key words
- critical values
- the roots where the expression is 0
- between
- one inequality, like a ≤ x ≤ b
- outside
- two parts, like x < a or x > b
Practise this
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