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Chapter 3: Problem 82

What is a rational inequality?

Short Answer

Expert verified
A rational inequality is an inequality that contains a rational expression. Rational expressions are fractions where the numerator and/or the denominator are polynomials. Solving this type of inequality requires finding the critical numbers and testing intervals. Our real solutions are intervals that satisfy the inequality.

Step by step solution

01

Define Rational Inequality

A rational inequality is an inequality that contains a rational expression, a fraction in which the numerator and/or the denominator are polynomials.
02

Point Out Essential Characteristics

These inequalities deal with fractions where the numerator and/or the denominator are polynomials. Like simple fractions, infix rules apply: we cannot divide by zero, which makes understanding undefined values essential.
03

Explain Solving Method

To solve a rational inequality: 1. Simplify the rational expression if it's not already in reduced form. 2. Find the critical numbers by setting the numerator equal to zero, and the denominator as well. (Regarding the denominator, we are looking for values that make the denominator zero as we cannot divide by zero.) 3. Identify the intervals formed by these critical numbers, and test numbers within these intervals in the original inequality.
04

Apply the Concepts

As an example, let's say we have the inequality \(\frac{x-1}{x+2} \geq 0\). The first step would be to find the critical numbers, which are \(x = 1\) and \(x = -2\). We pick a test number from each interval \(-\infty, -2\), \(-2, 1\), and \(1, \infty\) and substitute these numbers back into our inequality. From this, we can find the solution to our inequality.

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