Solving Rational Inequalities: A Step-by-Step Guide

Rational inequalities are a type of inequality involving a rational expression, which is a fraction where the numerator and denominator are polynomials. Solving rational inequalities is similar to solving polynomial inequalities, but with an extra layer of complexity due to the denominator. This guide will walk you through the process step-by-step, making it clear and understandable.

Understanding the Basics

Before diving into the steps, let’s define some key terms:

• **Rational Expression:** A fraction where the numerator and denominator are polynomials, such as (x^2 + 1) / (x – 2).
• **Critical Numbers:** Values of the variable that make the rational expression equal to zero or undefined. These numbers divide the number line into intervals where the expression is either positive or negative.
• **Sign Chart:** A visual representation of the sign of the rational expression in different intervals determined by the critical numbers.

Steps to Solve Rational Inequalities

Here’s a step-by-step guide to solving rational inequalities:

1. **Find the Critical Numbers:**
   • Set the numerator equal to zero and solve for x.
   • Set the denominator equal to zero and solve for x.
   • These solutions are your critical numbers. They represent the points where the rational expression changes sign.
2. **Create a Sign Chart:**
   • Draw a number line and mark the critical numbers on it.
   • The critical numbers divide the number line into intervals.
   • Choose a test value within each interval and evaluate the rational expression at that value.
   • The sign of the expression in each interval will determine whether it’s positive or negative.
3. **Determine the Solution:**
   • Identify the intervals where the rational expression satisfies the inequality.
   • Consider any restrictions on the domain (values that make the denominator zero). These values are excluded from the solution.
   • Write the solution in interval notation, remembering to include or exclude endpoints based on whether the inequality is strict or not.

Example

Let’s solve the rational inequality (x + 2) / (x – 1) > 0

1. **Find the Critical Numbers:**
   • Numerator: x + 2 = 0 => x = -2
   • Denominator: x – 1 = 0 => x = 1
   • Critical numbers: x = -2, x = 1
2. **Create a Sign Chart:**
   

   Interval	x < -2	-2 < x < 1	x > 1
   Test Value	x = -3	x = 0	x = 2
   (x + 2) / (x – 1)	+ / – = –	+ / – = –	+ / + = +

3. **Determine the Solution:**
   • The expression is positive in the interval x > 1.
   • The denominator cannot be zero, so x = 1 is excluded.
   • The solution is x ∈ (1, ∞)

Conclusion

Solving rational inequalities requires a systematic approach. By understanding the steps and using a sign chart, you can effectively determine the solution set for any given inequality. Remember to consider the restrictions imposed by the denominator and to express your solution in interval notation.