Solving Rational Inequalities with Sign Charts

Rational inequalities are inequalities that involve rational expressions, which are fractions where the numerator and denominator are polynomials. Solving these inequalities can be a bit more complex than solving linear or quadratic inequalities, but with the right approach, it becomes manageable. One of the most common and effective methods is using sign charts.

Understanding Sign Charts

A sign chart is a visual tool that helps determine the intervals where a function is positive, negative, or zero. It's particularly useful for solving inequalities involving rational expressions because it allows us to track the sign of the expression across different intervals.

Steps to Solve Rational Inequalities using Sign Charts

1. Find the Critical Numbers: Critical numbers are the values of the variable that make the numerator or denominator of the rational expression equal to zero. These numbers divide the number line into intervals.
2. Create a Sign Chart: Draw a number line and mark the critical numbers on it. These numbers will divide the number line into intervals. Above the number line, write the numerator and denominator of the rational expression. Below the number line, create a row to represent the sign of the expression in each interval.
3. Determine the Sign of Each Factor: For each interval, choose a test value within the interval and substitute it into the numerator and denominator of the rational expression. Determine the sign of each factor (numerator and denominator) for that test value.
4. Determine the Sign of the Expression: The sign of the entire rational expression in each interval is determined by the product of the signs of the numerator and denominator. If the signs are the same (both positive or both negative), the expression is positive. If the signs are different, the expression is negative.
5. Identify the Solution Intervals: Based on the sign chart, identify the intervals where the rational expression satisfies the given inequality. Remember to consider whether the inequality includes equality (≤ or ≥) and exclude any critical numbers that make the denominator zero.

Example: Solving a Rational Inequality

Let's solve the inequality:

(x + 2) / (x - 1) ≤ 0

1. Find the Critical Numbers:
   * The numerator is zero when x = -2.
   * The denominator is zero when x = 1.
   * Critical numbers: -2 and 1
2. Create a Sign Chart:
   ```
   -2 1
   --- ---
   x + 2
   x - 1
   (x + 2) / (x - 1)
   ```
3. Determine the Sign of Each Factor:
   * Interval (-∞, -2): Choose x = -3. (x + 2) is negative, (x - 1) is negative, so (x + 2) / (x - 1) is positive.
   * Interval (-2, 1): Choose x = 0. (x + 2) is positive, (x - 1) is negative, so (x + 2) / (x - 1) is negative.
   * Interval (1, ∞): Choose x = 2. (x + 2) is positive, (x - 1) is positive, so (x + 2) / (x - 1) is positive.
4. Determine the Sign of the Expression:
   ```
   -2 1
   --- ---
   x + 2 - + +
   x - 1 - - +
   (x + 2) / (x - 1) + - +
   ```
5. Identify the Solution Intervals:
   The inequality is ≤ 0, which means we want intervals where the expression is negative or zero. The solution is (-2, 1]. Note that we include -2 because the inequality includes equality, but we exclude 1 because it makes the denominator zero.

Summary

Sign charts provide a clear and organized way to solve rational inequalities. By understanding the steps involved and practicing with examples, you can confidently solve these types of problems. Remember to always check for excluded values that make the denominator zero.