Solving Rational Inequalities: A Step-by-Step Guide

Rational inequalities involve expressions where variables appear in both the numerator and denominator. Solving these inequalities requires a slightly different approach compared to solving linear or quadratic inequalities. This guide will walk you through the steps involved in solving rational inequalities using sign charts. We’ll break down the process into manageable steps and illustrate it with examples.

Understanding Rational Inequalities

A rational inequality is an inequality that involves a rational expression. A rational expression is a fraction where the numerator and denominator are polynomials. Here are a few examples of rational inequalities:

• x/(x + 2) > 0

• (x – 1)/(x + 3) < 2

• (x^2 – 4)/(x – 1) ≥ 0

Steps to Solve Rational Inequalities

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

1. Find the Critical Numbers

   Critical numbers are the values of the variable that make either the numerator or the denominator of the rational expression equal to zero. These numbers are important because they divide the number line into intervals where the expression’s sign can change.

   To find the critical numbers:

   1. Set the numerator equal to zero and solve for the variable.

   2. Set the denominator equal to zero and solve for the variable.

2. Create a Sign Chart

   A sign chart helps visualize the intervals where the expression is positive, negative, or zero. Here’s how to create a sign chart:

   1. Draw a number line and mark the critical numbers on it.

   2. Choose a test value from each interval (between the critical numbers) and substitute it into the original rational expression.

   3. Determine the sign of the expression for each interval and mark it on the sign chart.

3. Interpret the Results

   Once the sign chart is complete, you can identify the intervals that satisfy the original inequality. Consider the following:

   • If the inequality is > 0 or ≥ 0, look for intervals where the expression is positive.

   • If the inequality is < 0 or ≤ 0, look for intervals where the expression is negative.

   Remember to exclude any critical numbers that make the denominator zero, as they are not part of the solution set.

Example: Solving a Rational Inequality

Let’s solve the following rational inequality:

(x – 1)/(x + 2) > 0

1. Find the Critical Numbers

   Set the numerator equal to zero:

   x – 1 = 0

   x = 1

   Set the denominator equal to zero:

   x + 2 = 0

   x = -2

   So, the critical numbers are x = 1 and x = -2.

2. Create a Sign Chart

   Draw a number line and mark the critical numbers:

   |—|—|—|—|

   -2 1

   Choose test values from each interval:

   • Interval 1 (x < -2): Test value x = -3. (-3 – 1)/(-3 + 2) = 4 > 0

   • Interval 2 (-2 < x < 1): Test value x = 0. (0 – 1)/(0 + 2) = -1/2 < 0

   • Interval 3 (x > 1): Test value x = 2. (2 – 1)/(2 + 2) = 1/4 > 0

   Mark the signs on the sign chart:

   |—|—|—|—|

   -2 1

   + – +

3. Interpret the Results

   The inequality is > 0, so we look for intervals where the expression is positive. The solution set is:

   x < -2 or x > 1

Key Points to Remember

• Always exclude critical numbers that make the denominator zero.

• The sign chart helps visualize the intervals where the expression is positive, negative, or zero.

• Interpret the results based on the inequality symbol (>, <, ≥, ≤).

Conclusion

Solving rational inequalities involves finding critical numbers, creating a sign chart, and interpreting the results. By following these steps, you can effectively solve rational inequalities and understand the solution sets. Remember to practice with different examples to solidify your understanding of the process.