Polynomial and rational inequalities, and systems | Algebra 2

Same tools, bigger expressions

Everything in this lesson combines two tools you already have:

the sign chart for products and quotients;
the sign of a quadratic (negative between its roots, positive outside, when $$a > 0$$ ).

The plan is always the same: factor completely, find every critical value, and track the sign of each factor across the number line.

Polynomial inequalities

ExampleSolve $$(x^2 - 9)(x + 1) > 0$$

Factor completely: $$(x - 3)(x + 3)(x + 1)$$ . Critical values: $$-3$$ , $$-1$$ , $$3$$ .

Factor	$$x < -3$$	$$-3 < x < -1$$	$$-1 < x < 3$$	$$x > 3$$
$$x - 3$$	$$-$$	$$-$$	$$-$$	$$+$$
$$x + 3$$	$$-$$	$$+$$	$$+$$	$$+$$
$$x + 1$$	$$-$$	$$-$$	$$+$$	$$+$$
product	$$-$$	$$+$$	$$-$$	$$+$$

-3 < x < -1 \quad \text{or} \quad x > 3

ExampleSolve $x^3 - 4x^2 + 3x \le 0$

x(x^2 - 4x + 3) = x(x - 1)(x - 3)

Factor	$$x < 0$$	$$0 < x < 1$$	$$1 < x < 3$$	$$x > 3$$
$$x$$	$$-$$	$$+$$	$$+$$	$$+$$
$$x - 1$$	$$-$$	$$-$$	$$+$$	$$+$$
$$x - 3$$	$$-$$	$$-$$	$$-$$	$$+$$
product	$$-$$	$$+$$	$$-$$	$$+$$

Negative or zero, endpoints included:

x \le 0 \quad \text{or} \quad 1 \le x \le 3 \qquad (-\infty, 0] \cup [1, 3]

Rational inequalities

ExampleSolve $\frac{x^2 - x - 6}{x + 4} < 0$

Factor the numerator: $$(x - 3)(x + 2)$$ . Critical values: $$-4$$ , $$-2$$ , $$3$$ .

Factor	$$x < -4$$	$$-4 < x < -2$$	$$-2 < x < 3$$	$$x > 3$$
$$x - 3$$	$$-$$	$$-$$	$$-$$	$$+$$
$$x + 2$$	$$-$$	$$-$$	$$+$$	$$+$$
$$x + 4$$	$$-$$	$$+$$	$$+$$	$$+$$
quotient	$$-$$	$$+$$	$$-$$	$$+$$

x < -4 \quad \text{or} \quad -2 < x < 3

Systems of inequalities

In a system, every inequality must hold at the same time. Solve each one separately, then keep the overlap, just like an "and" compound inequality.

Practice

Solve $\frac{x + 2}{x^2 - 1} \ge 0$ .

ShowHide the solution

Critical values: $$-2$$ (numerator), $$-1$$ and $$1$$ (denominator).

Factor	$$x < -2$$	$$-2 < x < -1$$	$$-1 < x < 1$$	$$x > 1$$
$$x + 2$$	$$-$$	$$+$$	$$+$$	$$+$$
$$x^2 - 1$$	$$+$$	$$+$$	$$-$$	$$+$$
quotient	$$-$$	$$+$$	$$-$$	$$+$$

$$-2$$ is included (numerator zero); $\pm 1$ are excluded.

-2 \le x < -1 \quad \text{or} \quad x > 1

Quick check

What is the solution of $\frac{x^2}{x^2 + 1} \ge 0$ ?

Quick check

What is the solution of $$(x - 1)^2(x + 2) < 0$$ ?

Factor	$\(x < -3\)$	$\(-3 < x < -1\)$	$\(-1 < x < 3\)$	$\(x > 3\)$
$\(x - 3\)$	$\(-\)$	$\(-\)$	$\(-\)$	$\(+\)$
$\(x + 3\)$	$\(-\)$	$\(+\)$	$\(+\)$	$\(+\)$
$\(x + 1\)$	$\(-\)$	$\(-\)$	$\(+\)$	$\(+\)$
product	$\(-\)$	$\(+\)$	$\(-\)$	$\(+\)$

Factor	$\(x < 0\)$	$\(0 < x < 1\)$	$\(1 < x < 3\)$	$\(x > 3\)$
$\(x\)$	$\(-\)$	$\(+\)$	$\(+\)$	$\(+\)$
$\(x - 1\)$	$\(-\)$	$\(-\)$	$\(+\)$	$\(+\)$
$\(x - 3\)$	$\(-\)$	$\(-\)$	$\(-\)$	$\(+\)$
product	$\(-\)$	$\(+\)$	$\(-\)$	$\(+\)$

Factor	$\(x < -4\)$	$\(-4 < x < -2\)$	$\(-2 < x < 3\)$	$\(x > 3\)$
$\(x - 3\)$	$\(-\)$	$\(-\)$	$\(-\)$	$\(+\)$
$\(x + 2\)$	$\(-\)$	$\(-\)$	$\(+\)$	$\(+\)$
$\(x + 4\)$	$\(-\)$	$\(+\)$	$\(+\)$	$\(+\)$
quotient	$\(-\)$	$\(+\)$	$\(-\)$	$\(+\)$

Factor	$\(x < -2\)$	$\(-2 < x < -1\)$	$\(-1 < x < 1\)$	$\(x > 1\)$
$\(x + 2\)$	$\(-\)$	$\(+\)$	$\(+\)$	$\(+\)$
$\(x^2 - 1\)$	$\(+\)$	$\(+\)$	$\(-\)$	$\(+\)$
quotient	$\(-\)$	$\(+\)$	$\(-\)$	$\(+\)$