### Solving Inequalities #### Problem Statement: Find the solution set of the inequality: \[ \frac{x - 4}{x - 6} \leq \frac{x}{x - 1} \] Express your answer in interval notation. #### Multiple Choice Answers: 1. \((- \infty, -4] \cup (1, 6)\) 2. \((- \infty, -4] \cup (6, \infty)\) 3. \([-4, 1) \cup (6, \infty)\) 4. \((- \infty, -4) \cup (1, 6)\) 5. \((-4, 1) \cup (6, \infty)\) ### Explanation: To solve the inequality \(\frac{x - 4}{x - 6} \leq \frac{x}{x - 1}\), we want to combine the results of algebraic manipulation and testing intervals to find where the inequality holds true. Detailed steps to solve the inequality are as follows: 1. **Find common denominators and combine terms**: \[ \frac{x - 4}{x - 6} - \frac{x}{x - 1} \leq 0 \] 2. **Combine the fractions:** - Establish a common denominator and perform the subtraction to have a single inequality expression. - Simplify the combined fraction. 3. **Find critical points where either the numerator or denominator is zero:** - Identify the values of \(x\) that make the numerator or denominator zero. These are potential boundary points for your intervals. 4. **Test intervals around the critical points:** - Determine the sign of the inequality in the intervals defined by the critical points. 5. **Combine the valid intervals:** - From the testing, combine the intervals where the inequality holds true. Remember to carefully handle the boundaries of intervals because of the inequality being \(\leq\). ### Summary and Conclusion: After working through the steps and testing the intervals, choose the correct interval notation that matches the solution found from solving the inequality.
### Solving Inequalities #### Problem Statement: Find the solution set of the inequality: \[ \frac{x - 4}{x - 6} \leq \frac{x}{x - 1} \] Express your answer in interval notation. #### Multiple Choice Answers: 1. \((- \infty, -4] \cup (1, 6)\) 2. \((- \infty, -4] \cup (6, \infty)\) 3. \([-4, 1) \cup (6, \infty)\) 4. \((- \infty, -4) \cup (1, 6)\) 5. \((-4, 1) \cup (6, \infty)\) ### Explanation: To solve the inequality \(\frac{x - 4}{x - 6} \leq \frac{x}{x - 1}\), we want to combine the results of algebraic manipulation and testing intervals to find where the inequality holds true. Detailed steps to solve the inequality are as follows: 1. **Find common denominators and combine terms**: \[ \frac{x - 4}{x - 6} - \frac{x}{x - 1} \leq 0 \] 2. **Combine the fractions:** - Establish a common denominator and perform the subtraction to have a single inequality expression. - Simplify the combined fraction. 3. **Find critical points where either the numerator or denominator is zero:** - Identify the values of \(x\) that make the numerator or denominator zero. These are potential boundary points for your intervals. 4. **Test intervals around the critical points:** - Determine the sign of the inequality in the intervals defined by the critical points. 5. **Combine the valid intervals:** - From the testing, combine the intervals where the inequality holds true. Remember to carefully handle the boundaries of intervals because of the inequality being \(\leq\). ### Summary and Conclusion: After working through the steps and testing the intervals, choose the correct interval notation that matches the solution found from solving the inequality.
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
![### Solving Inequalities
#### Problem Statement:
Find the solution set of the inequality:
\[
\frac{x - 4}{x - 6} \leq \frac{x}{x - 1}
\]
Express your answer in interval notation.
#### Multiple Choice Answers:
1. \((- \infty, -4] \cup (1, 6)\)
2. \((- \infty, -4] \cup (6, \infty)\)
3. \([-4, 1) \cup (6, \infty)\)
4. \((- \infty, -4) \cup (1, 6)\)
5. \((-4, 1) \cup (6, \infty)\)
### Explanation:
To solve the inequality \(\frac{x - 4}{x - 6} \leq \frac{x}{x - 1}\), we want to combine the results of algebraic manipulation and testing intervals to find where the inequality holds true. Detailed steps to solve the inequality are as follows:
1. **Find common denominators and combine terms**:
\[
\frac{x - 4}{x - 6} - \frac{x}{x - 1} \leq 0
\]
2. **Combine the fractions:**
- Establish a common denominator and perform the subtraction to have a single inequality expression.
- Simplify the combined fraction.
3. **Find critical points where either the numerator or denominator is zero:**
- Identify the values of \(x\) that make the numerator or denominator zero. These are potential boundary points for your intervals.
4. **Test intervals around the critical points:**
- Determine the sign of the inequality in the intervals defined by the critical points.
5. **Combine the valid intervals:**
- From the testing, combine the intervals where the inequality holds true.
Remember to carefully handle the boundaries of intervals because of the inequality being \(\leq\).
### Summary and Conclusion:
After working through the steps and testing the intervals, choose the correct interval notation that matches the solution found from solving the inequality.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dca08d-d7c1-4511-81d7-e3b17171b463%2F3605cd36-4064-4f08-8740-a970f0e4faa5%2Fm2c2ggq_processed.png&w=3840&q=75)
Transcribed Image Text:### Solving Inequalities
#### Problem Statement:
Find the solution set of the inequality:
\[
\frac{x - 4}{x - 6} \leq \frac{x}{x - 1}
\]
Express your answer in interval notation.
#### Multiple Choice Answers:
1. \((- \infty, -4] \cup (1, 6)\)
2. \((- \infty, -4] \cup (6, \infty)\)
3. \([-4, 1) \cup (6, \infty)\)
4. \((- \infty, -4) \cup (1, 6)\)
5. \((-4, 1) \cup (6, \infty)\)
### Explanation:
To solve the inequality \(\frac{x - 4}{x - 6} \leq \frac{x}{x - 1}\), we want to combine the results of algebraic manipulation and testing intervals to find where the inequality holds true. Detailed steps to solve the inequality are as follows:
1. **Find common denominators and combine terms**:
\[
\frac{x - 4}{x - 6} - \frac{x}{x - 1} \leq 0
\]
2. **Combine the fractions:**
- Establish a common denominator and perform the subtraction to have a single inequality expression.
- Simplify the combined fraction.
3. **Find critical points where either the numerator or denominator is zero:**
- Identify the values of \(x\) that make the numerator or denominator zero. These are potential boundary points for your intervals.
4. **Test intervals around the critical points:**
- Determine the sign of the inequality in the intervals defined by the critical points.
5. **Combine the valid intervals:**
- From the testing, combine the intervals where the inequality holds true.
Remember to carefully handle the boundaries of intervals because of the inequality being \(\leq\).
### Summary and Conclusion:
After working through the steps and testing the intervals, choose the correct interval notation that matches the solution found from solving the inequality.
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