Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Solve the Rational Inequality**
\[ \frac{2}{x-4} > \frac{x+1}{x-2} \]
To solve this rational inequality, perform the following steps:
1. **Identify Critical Points:**
- Set the denominators equal to zero to find the points where the rational expressions are undefined:
- \(x - 4 = 0 \Rightarrow x = 4\)
- \(x - 2 = 0 \Rightarrow x = 2\)
2. **Find the Common Denominator:**
- Multiply both sides by \((x-4)(x-2)\) to eliminate the fractions, keeping in mind the critical points.
3. **Solve the Inequality:**
- Simplify and solve the inequality:
\[ 2(x-2) > (x+1)(x-4) \]
4. **Determine the Sign:**
- Analyze the intervals defined by the critical points: \(x<2\), \(2<x<4\), and \(x>4\).
- Test a point from each interval in the simplified inequality to determine the sign.
5. **Combine Solutions:**
- Take note of the intervals that satisfy the inequality and consider any restrictions from the critical points.
Remember to check for any excluded values where the original inequality is undefined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6ad60fb3-d7c7-4aa0-8ac8-5f8ba0f10fe4%2F80403d59-acc6-45ea-8711-4f12b2af2ef8%2F5we5cr4_processed.png&w=3840&q=75)
Transcribed Image Text:**Solve the Rational Inequality**
\[ \frac{2}{x-4} > \frac{x+1}{x-2} \]
To solve this rational inequality, perform the following steps:
1. **Identify Critical Points:**
- Set the denominators equal to zero to find the points where the rational expressions are undefined:
- \(x - 4 = 0 \Rightarrow x = 4\)
- \(x - 2 = 0 \Rightarrow x = 2\)
2. **Find the Common Denominator:**
- Multiply both sides by \((x-4)(x-2)\) to eliminate the fractions, keeping in mind the critical points.
3. **Solve the Inequality:**
- Simplify and solve the inequality:
\[ 2(x-2) > (x+1)(x-4) \]
4. **Determine the Sign:**
- Analyze the intervals defined by the critical points: \(x<2\), \(2<x<4\), and \(x>4\).
- Test a point from each interval in the simplified inequality to determine the sign.
5. **Combine Solutions:**
- Take note of the intervals that satisfy the inequality and consider any restrictions from the critical points.
Remember to check for any excluded values where the original inequality is undefined.
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