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 seem daunting, but using sign charts provides a systematic and efficient approach. This method helps visualize the solution by breaking down the inequality into intervals based on critical numbers, where the expression equals zero or is undefined.

Step-by-Step Guide to Solving Rational Inequalities

1. Find Critical Numbers:
   • Set the numerator equal to zero and solve for x. These values are where the expression equals zero.
   • Set the denominator equal to zero and solve for x. These values are where the expression is undefined.
2. Create a Sign Chart:
   • List the critical numbers on a number line, separating it into intervals.
   • Choose a test value within each interval and substitute it into the original inequality.
   • Determine the sign (positive or negative) of the expression within each interval based on the test value.
3. Interpret the Results:
   • Identify the intervals where the inequality is true (satisfies the original inequality).
   • Write the solution set in interval notation, excluding any critical numbers that make the expression undefined.

Example: Solving a Rational Inequality

Let's solve the inequality: (x - 2) / (x + 1) > 0

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

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

3. Interpret the Results:

   The inequality is true (positive) when x < -1 or x > 2. Therefore, the solution set is:

   (-∞, -1) U (2, ∞)

Key Points to Remember

• Always exclude critical numbers that make the denominator zero from the solution set.
• Sign charts help visualize the solution by clearly showing the intervals where the expression is positive or negative.
• This method is applicable to various types of rational inequalities, making it a valuable tool for understanding and solving these mathematical problems.

By following these steps and using sign charts, you can confidently solve rational inequalities and gain a deeper understanding of their solutions.