Solve The Following Rational Inequality Calculator

This guide explains how to solve rational inequalities like (x-2)/(x+3) > 0 using our calculator. We'll cover the method, provide a worked example, and answer common questions.

How to Solve Rational Inequalities

Rational inequalities involve fractions with polynomials in the numerator and denominator. To solve them:

Find the critical points by setting the numerator and denominator equal to zero
Plot these points on a number line
Test intervals between critical points to determine where the inequality holds true
Consider undefined points where the denominator equals zero

Remember that the denominator cannot be zero, so these points are excluded from the solution.

Step-by-Step Solution Process

Here's the detailed method for solving rational inequalities:

Identify critical points: Solve numerator = 0 and denominator = 0 separately
Plot critical points: Mark these points on a number line
Determine test intervals: The critical points divide the number line into intervals
Test each interval: Choose a test point from each interval and determine if it satisfies the inequality
Exclude undefined points: Points where denominator = 0 are not part of the solution
Combine results: The solution is the union of all intervals where the inequality holds true

For inequality (P(x)/Q(x)) > 0: 1. Find x where P(x) = 0 and Q(x) = 0 2. Plot these points on a number line 3. Test intervals between critical points 4. Solution is where test points satisfy the inequality

Worked Example

Let's solve (x-2)/(x+3) > 0 step by step:

Example Problem

Solve (x-2)/(x+3) > 0

Step 1: Find Critical Points

Set numerator and denominator equal to zero:

Numerator: x - 2 = 0 → x = 2
Denominator: x + 3 = 0 → x = -3

Step 2: Plot Critical Points

The critical points divide the number line into three intervals:

x < -3
-3 < x < 2
x > 2

Step 3: Test Each Interval

Choose test points from each interval:

For x < -3: Test x = -4 → (-4-2)/(-4+3) = -6/-1 = 6 > 0 → True
For -3 < x < 2: Test x = 0 → (0-2)/(0+3) = -2/3 < 0 → False
For x > 2: Test x = 3 → (3-2)/(3+3) = 1/6 > 0 → True

Step 4: Combine Results

The inequality holds true in the intervals x < -3 and x > 2, but x cannot equal -3 (denominator zero).

Solution

The solution to (x-2)/(x+3) > 0 is:

x ∈ (-∞, -3) ∪ (2, ∞)

Frequently Asked Questions

What is a rational inequality?

A rational inequality is an inequality that contains a rational expression (fraction) with polynomials in the numerator and denominator.

How do I solve rational inequalities?

Find critical points by setting numerator and denominator to zero, plot them on a number line, test intervals between critical points, and combine the results.

What happens when the denominator is zero?

The expression is undefined when the denominator is zero, so these points are excluded from the solution.

Can I use the calculator for any rational inequality?

Yes, our calculator can solve any rational inequality in the form (ax+b)/(cx+d) > 0 or similar linear forms.