College Algebra

Polynomial and Rational Inequalities

Lesson

To solve a polynomial or rational inequality, the easiest method is a sign chart. Find the values of $x$ where the expression equals zero or is undefined; those are the boundaries. Between boundaries, the expression keeps a constant sign — so test one value per interval and you know everything.

The strategy:

Move everything to one side so it reads (stuff) $\square$ 0.
Factor.
Find the boundary values: zeros of factors (and any denominator zeros for rational inequalities).
Test one point in each interval.
Pick the intervals whose sign matches the inequality.

For the problems in this topic, we’ve chosen inequalities whose solution is a single bounded interval. You report the length of that interval (the larger boundary minus the smaller one). This forces you to identify both boundaries and confirm the solution is the bounded piece.

Worked example 1

x^2 - 4x < 0

Factor: $x(x - 4) < 0$ . Boundaries: 0 and 4. Test $x = 2$ : $4 - 8 = -4$ , which is negative — the inequality is true.

Solution: $0 < x < 4$ . Length $= 4 - 0 = 4$ .

Worked example 2 — rational

\frac{x + 1}{x - 2} \leq 0

Boundaries: $x = -1$ (numerator zero) and $x = 2$ (denominator zero — must be excluded). Test $x = 0$ : $\frac{1}{-2} = -\tfrac{1}{2}$ , which is $\leq 0$ — true.

Solution: $-1 \leq x < 2$ . Length $= 2 - (-1) = 3$ .

How to type your answer

Type the length of the solution interval — a single number. Use a slash for fractions. Examples: 4, 3, 3/2, 7/2.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x^2 < 9

Problem 2

x^2 \leq 16

Problem 3

(x - 1)(x - 5) < 0

Problem 4

(x + 2)(x - 3) \leq 0

Practice

Standard problems matching the lesson.

Problem 5

x^2 - 4x < 0

Problem 6

x^2 + 3x - 4 < 0

Problem 7

x^2 - 5x + 6 \leq 0

Problem 8

x^2 + 2x - 8 \leq 0

Problem 9

2x^2 - 5x + 2 \leq 0

Problem 10

x^2 - 6x + 5 < 0

Problem 11

(x + 1)(x - 7) < 0

Problem 12

x^2 - 1 \leq 0

Problem 13

x^2 \leq 25

Problem 14

3x^2 < 12

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 15

\dfrac{x + 1}{x - 2} \leq 0

Problem 16

\dfrac{x - 3}{x + 1} < 0

Problem 17

2x^2 + 5x - 3 \leq 0

Problem 18

x^2 - 7x + 12 < 0

Problem 19

\dfrac{x - 1}{x + 3} \leq 0

Problem 20

3x^2 - x - 2 < 0

Problem 21

\dfrac{2x + 1}{x - 3} \leq 0

Problem 22

x^2 - 2x - 8 < 0