Move all terms to one side, find the roots, then read the required intervals from a sketch.
Include roots for \(\le\) or \(\ge\). For a repeated root or no real roots, use the touching or non-crossing graph.
Two distinct roots: \(p<q\), \(a>0\)
| Inequality | Solution |
|---|---|
| \(ax^2+bx+c>0\) | |
| \(ax^2+bx+c\ge 0\) | |
| \(ax^2+bx+c<0\) | |
| \(ax^2+bx+c\le 0\) |
Common errors
| Inequality | Solution | Take note |
|---|---|---|
| \(x^2>9\) | \(x<-3\) or \(x>3\) | Include both intervals. |
| \(x^2>6x\) | \(x<0\) or \(x>6\) | Do not divide by \(x\); its sign is unknown. |
| \((x-2)(x-4)<0\) | \(2<x<4\) | Use the sign of the product. |
Multiplying or dividing by a negative number reverses the inequality sign.
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