Solve the inequality analytically. Make sure you show all your work on how you determined the intervals. 3 5 X X-2
Solve the inequality analytically. Make sure you show all your work on how you determined the intervals. 3 5 X X-2
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![**14) Solve the inequality analytically. Make sure you show all your work on how you determined the intervals.**
\[ \frac{3}{x} - \frac{5}{x - 2} \leq 0 \]
This problem requires finding the solution set for the given inequality by determining the intervals. Below is the step-by-step procedure to solve the inequality.
1. **Combine the fractions:** Find a common denominator for the left-hand side of the inequality.
\[ \frac{3(x - 2) - 5x}{x(x - 2)} \leq 0 \]
\[ \frac{3x - 6 - 5x}{x(x - 2)} \leq 0 \]
\[ \frac{-2x - 6}{x(x - 2)} \leq 0 \]
2. **Simplify the numerator:**
\[ \frac{-2(x + 3)}{x(x - 2)} \leq 0 \]
3. **Critical points:** Determine the critical points by setting the numerator and denominator equal to zero:
- Numerator: \(-2(x + 3) = 0\)
\[ x + 3 = 0 \rightarrow x = -3 \]
- Denominator: \(x(x - 2) = 0\)
\[ x = 0 \quad \text{or} \quad x = 2 \]
4. **Determine the intervals:** The critical points divide the number line into several intervals. Test each interval to determine where the inequality holds:
\[ (-\infty, -3), (-3, 0), (0, 2), (2, \infty) \]
5. **Test intervals:**
- Choose a test point from each interval and substitute it into the inequality to check if it holds true.
For \(x \in (-\infty, -3)\):
Test with \( x = -4 \):
\[ \frac{-2(-4 + 3)}{-4(-4 - 2)} = \frac{-2(-1)}{-4(-6)} = \frac{2}{24} > 0 \]
For \(x \in (-3, 0)\):
Test with \( x = -2 \):
\[ \frac{-2(-2 + 3)}{-2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc93a7fac-dada-46a7-8fba-a69354232c26%2F2ab5adc8-a075-4964-a234-2d36da98e5c0%2Felnqowv.jpeg&w=3840&q=75)
Transcribed Image Text:**14) Solve the inequality analytically. Make sure you show all your work on how you determined the intervals.**
\[ \frac{3}{x} - \frac{5}{x - 2} \leq 0 \]
This problem requires finding the solution set for the given inequality by determining the intervals. Below is the step-by-step procedure to solve the inequality.
1. **Combine the fractions:** Find a common denominator for the left-hand side of the inequality.
\[ \frac{3(x - 2) - 5x}{x(x - 2)} \leq 0 \]
\[ \frac{3x - 6 - 5x}{x(x - 2)} \leq 0 \]
\[ \frac{-2x - 6}{x(x - 2)} \leq 0 \]
2. **Simplify the numerator:**
\[ \frac{-2(x + 3)}{x(x - 2)} \leq 0 \]
3. **Critical points:** Determine the critical points by setting the numerator and denominator equal to zero:
- Numerator: \(-2(x + 3) = 0\)
\[ x + 3 = 0 \rightarrow x = -3 \]
- Denominator: \(x(x - 2) = 0\)
\[ x = 0 \quad \text{or} \quad x = 2 \]
4. **Determine the intervals:** The critical points divide the number line into several intervals. Test each interval to determine where the inequality holds:
\[ (-\infty, -3), (-3, 0), (0, 2), (2, \infty) \]
5. **Test intervals:**
- Choose a test point from each interval and substitute it into the inequality to check if it holds true.
For \(x \in (-\infty, -3)\):
Test with \( x = -4 \):
\[ \frac{-2(-4 + 3)}{-4(-4 - 2)} = \frac{-2(-1)}{-4(-6)} = \frac{2}{24} > 0 \]
For \(x \in (-3, 0)\):
Test with \( x = -2 \):
\[ \frac{-2(-2 + 3)}{-2
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