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Quadratic inequalities
Quadratic inequalities have answers that are either between two values or outside them. A quick sketch makes it clear.
Find the critical values
Solve the matching equation. For x2 − x − 6 < 0, solve x2 − x − 6 = 0: (x − 3)(x + 2) = 0, so x = 3 and x = −2.
Sketch
y = x2 − x − 6 is a U shape crossing at −2 and 3.
< 0 means below the x-axis: between the roots, −2 < x < 3.
> 0 means above: outside, x < −2 or x > 3.
< 0 means below the x-axis: between the roots, −2 < x < 3.
> 0 means above: outside, x < −2 or x > 3.
Regions
For linear inequalities in two variables, shade the side of each line that works. Test a point like (0, 0) to choose the side.
Solve x2 − 9 ≥ 0.
- Critical values: x = −3 and x = 3
- U shape; ≥ 0 means on or above the axis
- x ≤ −3 or x ≥ 3
Answer: x ≤ −3 or x ≥ 3
Solve the equation for the critical values, sketch the U-shaped curve, and read off where it is below (< 0) or above (> 0) the x-axis.
Check you have got it
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