Polynomial and Rational Inequalities

Key vocabulary and concepts for solving polynomial and rational inequalities using critical points and interval notation.

Polynomial inequality

An inequality involving a polynomial expression, such as $(x-3)(x+2)(x-1) \le 0$.

Rational inequality

An inequality with a fraction containing variables, such as $\frac{x+2}{x-1} \le 0$.

Real number line

A line used to represent all real numbers, extending from negative infinity ($-\infty$) to positive infinity ($\infty$).

Interval notation

A method to represent solution sets using parentheses $( \, )$ and brackets $[ \, ]$, for example $[-2, 1)$.

Brackets $[ \, ]$

Symbols used in interval notation to indicate that a value is included in the solution set.

Parentheses $( \, )$

Symbols used in interval notation to indicate that a value is not included in the solution set.

Critical points

Values found by setting each factor of an inequality equal to $0$, which serve to divide the real number line into test intervals.

Test intervals

Segments of the real number line created by zeros where a single number is tested to see if the expression is positive or negative in that range.

Denominator restriction

A rule in rational inequalities stating that a denominator can never equal $0$, which means those values are excluded from the solution set.

Solution set for $(x-3)(x+2)(x-1) \le 0$

$(-\infty, -2] \cup [1, 3]$, derived from critical points $-2$, $1$, and $3$.

Solution set for $\frac{x+2}{x-1} \le 0$

$[-2, 1)$, with the value $1$ excluded because it would make the denominator equal to $0$.