Precalculus / Unit 2: Polynomial and Rational Functions
2.6 Polynomial and Rational Inequalities
Notes
To solve $f(x) > 0$ or $f(x) < 0$, find the zeros (and undefined points), then test the intervals between them.
- Use a sign chart. Zeros of the numerator and denominator split the number line.
- Include zeros for $\le$ or $\ge$, but never include values that make a denominator zero.
- Example: $(x - 1)(x + 2) > 0$ has zeros at 1 and $-2$, and is positive outside them: $x < -2$ or $x > 1$.
Practice
Solve $(x - 1)(x + 2) > 0$.
The parabola opens up, so it is positive outside the zeros $-2$ and $1$.
Solve $x^2 - 4x - 5 \le 0$.
$(x - 5)(x + 1) \le 0$ is true between the zeros, including them.
Solve $\displaystyle \frac{x - 3}{x + 1} \le 0$.
The fraction is negative between $-1$ and $3$; $x = 3$ gives 0 (allowed) but $x = -1$ is undefined.
Ready for more? Take the Unit 2 test.