- Solve: (3x-1)(x²-4) 5x(x−7) 20
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°.
Related questions
Question
![## Solving Rational Inequalities
### Problem Statement
3. Solve the inequality:
\[
\frac{(3x-1)(x^2-4)}{5x(x-7)} \geq 0
\]
### Explanation
To solve this rational inequality, follow these steps:
1. **Factor the Expression**:
- Recognize that \(x^2 - 4\) can be factored as \((x - 2)(x + 2)\).
2. **Identify Critical Points**:
- The critical points are found by setting the numerator and denominator equal to zero:
- \(3x - 1 = 0\) gives \(x = \frac{1}{3}\).
- \(x^2 - 4 = 0\) gives \(x = 2\) and \(x = -2\).
- \(5x(x-7) = 0\) gives critical points \(x = 0\) and \(x = 7\).
3. **Sign Analysis**:
- Determine the sign of the expression in each interval defined by the critical points:
- Intervals include \((- \infty, -2)\), \((-2, 0)\), \((0, \frac{1}{3})\), \((\frac{1}{3}, 2)\), \((2, 7)\), and \((7, \infty)\).
- Analyze the sign of the expression in each interval by choosing test points.
4. **Solution Set**:
- Combine intervals where the expression is positive or zero, according to the inequality \(\geq 0\).
### Conclusion
Finally, present the solution set, taking into account that any values making the denominator zero are excluded from the solution. Summarize the intervals for \(x\) that satisfy the inequality.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4f7ec96-caf0-42ec-8bfa-720897921654%2F3ae01a42-a05e-4483-9881-4a8fd9589ac9%2Fbcml9o_processed.png&w=3840&q=75)
Transcribed Image Text:## Solving Rational Inequalities
### Problem Statement
3. Solve the inequality:
\[
\frac{(3x-1)(x^2-4)}{5x(x-7)} \geq 0
\]
### Explanation
To solve this rational inequality, follow these steps:
1. **Factor the Expression**:
- Recognize that \(x^2 - 4\) can be factored as \((x - 2)(x + 2)\).
2. **Identify Critical Points**:
- The critical points are found by setting the numerator and denominator equal to zero:
- \(3x - 1 = 0\) gives \(x = \frac{1}{3}\).
- \(x^2 - 4 = 0\) gives \(x = 2\) and \(x = -2\).
- \(5x(x-7) = 0\) gives critical points \(x = 0\) and \(x = 7\).
3. **Sign Analysis**:
- Determine the sign of the expression in each interval defined by the critical points:
- Intervals include \((- \infty, -2)\), \((-2, 0)\), \((0, \frac{1}{3})\), \((\frac{1}{3}, 2)\), \((2, 7)\), and \((7, \infty)\).
- Analyze the sign of the expression in each interval by choosing test points.
4. **Solution Set**:
- Combine intervals where the expression is positive or zero, according to the inequality \(\geq 0\).
### Conclusion
Finally, present the solution set, taking into account that any values making the denominator zero are excluded from the solution. Summarize the intervals for \(x\) that satisfy the inequality.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning