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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![**Solve:**
\[
\frac{(3x - 1)(x^2 - 4)}{5x(x - 7)} \geq 0
\]
**Explanation:**
This is a rational inequality where the expression \(\frac{(3x - 1)(x^2 - 4)}{5x(x - 7)}\) needs to be solved for when it is greater than or equal to zero.
### Steps to Solve the Inequality:
1. **Factor the Quadratic:**
- The term \(x^2 - 4\) is a difference of squares and can be factored as \((x-2)(x+2)\).
2. **Identify Critical Points:**
- Set each factor in the numerator \((3x-1)(x-2)(x+2)\) and denominator \(5x(x-7)\) equal to zero to find critical points.
- \(3x - 1 = 0 \Rightarrow x = \frac{1}{3}\)
- \(x - 2 = 0 \Rightarrow x = 2\)
- \(x + 2 = 0 \Rightarrow x = -2\)
- \(5x = 0 \Rightarrow x = 0\)
- \(x - 7 = 0 \Rightarrow x = 7\)
3. **Plot the Critical Points on a Number Line:**
- The number line should include critical points: \(-2, 0, \frac{1}{3}, 2, 7\).
4. **Test Intervals:**
- Choose test points in each interval to determine whether the expression is positive or negative in those intervals.
- Check the sign of the expression in the intervals: \((-\infty, -2)\), \((-2, 0)\), \((0, \frac{1}{3})\), \((\frac{1}{3}, 2)\), \((2, 7)\), \((7, \infty)\).
5. **Consider Solutions:**
- Since the inequality is \(\geq 0\), include points where the expression is positive or zero.
- Exclude points where the denominator is zero because division by zero is undefined.
By following these steps, you can determine the solution set for](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4a55658-f75f-4303-9793-a2840126e3d4%2F95d508ff-513e-4c96-99f3-535af70962be%2Fnugiyx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Solve:**
\[
\frac{(3x - 1)(x^2 - 4)}{5x(x - 7)} \geq 0
\]
**Explanation:**
This is a rational inequality where the expression \(\frac{(3x - 1)(x^2 - 4)}{5x(x - 7)}\) needs to be solved for when it is greater than or equal to zero.
### Steps to Solve the Inequality:
1. **Factor the Quadratic:**
- The term \(x^2 - 4\) is a difference of squares and can be factored as \((x-2)(x+2)\).
2. **Identify Critical Points:**
- Set each factor in the numerator \((3x-1)(x-2)(x+2)\) and denominator \(5x(x-7)\) equal to zero to find critical points.
- \(3x - 1 = 0 \Rightarrow x = \frac{1}{3}\)
- \(x - 2 = 0 \Rightarrow x = 2\)
- \(x + 2 = 0 \Rightarrow x = -2\)
- \(5x = 0 \Rightarrow x = 0\)
- \(x - 7 = 0 \Rightarrow x = 7\)
3. **Plot the Critical Points on a Number Line:**
- The number line should include critical points: \(-2, 0, \frac{1}{3}, 2, 7\).
4. **Test Intervals:**
- Choose test points in each interval to determine whether the expression is positive or negative in those intervals.
- Check the sign of the expression in the intervals: \((-\infty, -2)\), \((-2, 0)\), \((0, \frac{1}{3})\), \((\frac{1}{3}, 2)\), \((2, 7)\), \((7, \infty)\).
5. **Consider Solutions:**
- Since the inequality is \(\geq 0\), include points where the expression is positive or zero.
- Exclude points where the denominator is zero because division by zero is undefined.
By following these steps, you can determine the solution set for
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