Inequalities

Method

1. Solve the corresponding equation $= 0$.
2. Sketch the parabola.
3. Read off the solution from the graph.

Example

$x^2 - 5x + 6 > 0$ → $(x-2)(x-3) > 0$

Roots at 2 and 3. Parabola opens upward → positive outside roots.

Solution: $x < 2$ or $x > 3$.

For $< 0$: between the roots.

Never multiply both sides by an expression that might be negative. Instead:

1. Rearrange to one side.
2. Find critical values (zeros of numerator and denominator).
3. Test intervals.

Example

$\frac{x+1}{x-2} > 0$

Critical values: $x = -1, x = 2$.

Test: $x < -1$: $\frac{(-)}{(-)} > 0$ ✓. $-1 < x < 2$: $\frac{(+)}{(-)} < 0$ ✗. $x > 2$: $\frac{(+)}{(+)} > 0$ ✓.

Solution: $x < -1$ or $x > 2$.

$\{x : x < 2\} \cup \{x : x > 3\}$

or: $x \in (-\infty, 2) \cup (3, \infty)$