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- Linear and quadratic inequalities
Linear and quadratic inequalities
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Inequalities are solved like equations, with two extra rules: flip the sign when you multiply or divide by a negative, and sketch quadratics.
Linear inequalities
Collect terms like an equation.
5x − 3 > 2x + 9
3x > 12, so x > 4
If you multiply or divide by a negative number, reverse the sign.
5x − 3 > 2x + 9
3x > 12, so x > 4
If you multiply or divide by a negative number, reverse the sign.
Quadratic inequalities
Rearrange so one side is 0, find the critical values, then sketch.
x2 − 5x − 14 < 0
(x − 7)(x + 2) < 0, critical values −2 and 7
The U-shaped graph is below the x-axis between them: −2 < x < 7
x2 − 5x − 14 < 0
(x − 7)(x + 2) < 0, critical values −2 and 7
The U-shaped graph is below the x-axis between them: −2 < x < 7
Above the axis
x2 − 9 ≥ 0: critical values ±3
The graph is on or above the axis outside them: x ≤ −3 or x ≥ 3
Both sides with x2 terms? Collect everything on one side first.
The graph is on or above the axis outside them: x ≤ −3 or x ≥ 3
Both sides with x2 terms? Collect everything on one side first.
Solve 2x2 + 3 > x2 + 4x.
- x2 − 4x + 3 > 0
- (x − 1)(x − 3) > 0, critical values 1 and 3
- The graph is above the axis outside the roots
Answer: x < 1 or x > 3
Find the critical values, sketch the parabola, and read off where it is above (> 0) or below (< 0) the x-axis.
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