Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.9: Dividing Real Numbers
Problem 10MRE
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![**Problem:** Solve each inequality algebraically.
**Inequality:** \( x^3 - x \geq 0 \)
**Solution:** This inequality can be factored and solved using the following steps:
1. **Factor the Expression:**
Factor \( x^3 - x \) by taking out the common factor \( x \):
\[
x^3 - x = x(x^2 - 1)
\]
Recognize that \( x^2 - 1 \) can be further factored as a difference of squares:
\[
x(x^2 - 1) = x(x - 1)(x + 1)
\]
2. **Determine Critical Points:**
Set each factor equal to zero to find critical points:
\[
x = 0, \quad x - 1 = 0 \quad \Rightarrow \quad x = 1, \quad x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
The critical points are \( x = -1, 0, 1 \).
3. **Test Intervals:**
Test intervals determined by critical points in the inequality \( x(x - 1)(x + 1) \geq 0 \):
- Interval \( (-\infty, -1) \)
- Interval \( (-1, 0) \)
- Interval \( (0, 1) \)
- Interval \( (1, \infty) \)
4. **Analyze Signs in Each Interval:**
- \( x \in (-\infty, -1) \): Choose \( x = -2 \), then \((-2)(-3)(-1) < 0\)
- \( x \in (-1, 0) \): Choose \( x = -0.5 \), then \((-0.5)(-1.5)(0.5) > 0\)
- \( x \in (0, 1) \): Choose \( x = 0.5 \), then \((0.5)(-0.5)(1.5) < 0\)
- \( x \in (1, \infty) \): Choose \( x = 2 \), then \((2)(1)(3) > 0\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb0e73cc8-5bc3-4bfd-b6e9-068f811e0db9%2F56faa28f-8899-4956-85e0-8e09d6bca9f8%2Fecjhfiu_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:** Solve each inequality algebraically.
**Inequality:** \( x^3 - x \geq 0 \)
**Solution:** This inequality can be factored and solved using the following steps:
1. **Factor the Expression:**
Factor \( x^3 - x \) by taking out the common factor \( x \):
\[
x^3 - x = x(x^2 - 1)
\]
Recognize that \( x^2 - 1 \) can be further factored as a difference of squares:
\[
x(x^2 - 1) = x(x - 1)(x + 1)
\]
2. **Determine Critical Points:**
Set each factor equal to zero to find critical points:
\[
x = 0, \quad x - 1 = 0 \quad \Rightarrow \quad x = 1, \quad x + 1 = 0 \quad \Rightarrow \quad x = -1
\]
The critical points are \( x = -1, 0, 1 \).
3. **Test Intervals:**
Test intervals determined by critical points in the inequality \( x(x - 1)(x + 1) \geq 0 \):
- Interval \( (-\infty, -1) \)
- Interval \( (-1, 0) \)
- Interval \( (0, 1) \)
- Interval \( (1, \infty) \)
4. **Analyze Signs in Each Interval:**
- \( x \in (-\infty, -1) \): Choose \( x = -2 \), then \((-2)(-3)(-1) < 0\)
- \( x \in (-1, 0) \): Choose \( x = -0.5 \), then \((-0.5)(-1.5)(0.5) > 0\)
- \( x \in (0, 1) \): Choose \( x = 0.5 \), then \((0.5)(-0.5)(1.5) < 0\)
- \( x \in (1, \infty) \): Choose \( x = 2 \), then \((2)(1)(3) > 0\)
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