Rational Inequalities

Front

What is a rational inequality?

Back

A rational inequality is an inequality that involves a rational expression, which is a fraction where the numerator and denominator are polynomials.

Front

How do you solve a rational inequality?

Back

To solve a rational inequality, first set the rational expression to zero, find the critical points, test intervals between these points, and determine where the inequality holds true.

Front

What are critical points in the context of rational inequalities?

Back

Critical points are the values of the variable that make the numerator or denominator of the rational expression zero.

Front

What does it mean for a rational inequality to be undefined?

Back

A rational inequality is undefined at points where the denominator equals zero, as division by zero is not possible.

Front

What is the difference between strict and non-strict inequalities?

Back

Strict inequalities (<, >) do not include the boundary points, while non-strict inequalities (≤, ≥) include the boundary points.

Front

How do you represent the solution set of a rational inequality?

Back

The solution set can be represented using interval notation, which indicates the ranges of values that satisfy the inequality.

Front

What is interval notation?

Back

Interval notation is a way of writing subsets of the real number line using parentheses and brackets to indicate whether endpoints are included or excluded.

Front

Give an example of a rational inequality.

Back

An example of a rational inequality is (x-2)/(x+3) < 0.

Front

What is the first step in solving the inequality (x-2)/(x+3) < 0?

Back

The first step is to find the critical points by setting the numerator and denominator to zero: x-2=0 (x=2) and x+3=0 (x=-3).

Front

How do you test intervals for a rational inequality?

Back

Choose test points from each interval created by the critical points and substitute them into the rational expression to see if the inequality holds.