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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![**Problem Statement:**
Find the solution of the following polynomial inequality. Express your answer in interval notation.
\[ x(x + 4)(x - 4) \geq 0 \]
**Solution Box:**
The solution involves identifying the intervals where the polynomial expression is greater than or equal to zero. Consider the critical points (roots) from the inequality: \( x = 0 \), \( x = -4 \), and \( x = 4 \).
1. **Test Intervals:** Break down the real number line into intervals based on the critical points:
- \((-∞, -4)\)
- \((-4, 0)\)
- \((0, 4)\)
- \((4, ∞)\)
2. **Sign Analysis:**
- For \( (-∞, -4) \), test with \( x = -5 \): the expression \((-5)(-1)(-9) = -45\) is negative.
- For \((-4, 0)\), test with \( x = -2 \): the expression \((-2)(2)(-6) = 24\) is positive.
- For \((0, 4)\), test with \( x = 2 \): the expression \((2)(6)(-2) = -24\) is negative.
- For \((4, ∞)\), test with \( x = 5 \): the expression \((5)(9)(1) = 45\) is positive.
3. **Include Critical Points:** Since the inequality is “greater than or equal to zero”, include points where the expression equals zero: \( x = -4 \), \( x = 0 \), \( x = 4 \).
4. **Solution in Interval Notation:**
The solution is \( [-4, 0] \cup [4, ∞) \).
Ensure each interval where the expression is non-negative is correctly identified and that edges of those intervals reflect equality conditions appropriately.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8111b40-f806-4f4f-8a96-c8bed7e93551%2F5494790a-f316-4dd1-b5a0-1dfa06233226%2Fe5pns7g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the solution of the following polynomial inequality. Express your answer in interval notation.
\[ x(x + 4)(x - 4) \geq 0 \]
**Solution Box:**
The solution involves identifying the intervals where the polynomial expression is greater than or equal to zero. Consider the critical points (roots) from the inequality: \( x = 0 \), \( x = -4 \), and \( x = 4 \).
1. **Test Intervals:** Break down the real number line into intervals based on the critical points:
- \((-∞, -4)\)
- \((-4, 0)\)
- \((0, 4)\)
- \((4, ∞)\)
2. **Sign Analysis:**
- For \( (-∞, -4) \), test with \( x = -5 \): the expression \((-5)(-1)(-9) = -45\) is negative.
- For \((-4, 0)\), test with \( x = -2 \): the expression \((-2)(2)(-6) = 24\) is positive.
- For \((0, 4)\), test with \( x = 2 \): the expression \((2)(6)(-2) = -24\) is negative.
- For \((4, ∞)\), test with \( x = 5 \): the expression \((5)(9)(1) = 45\) is positive.
3. **Include Critical Points:** Since the inequality is “greater than or equal to zero”, include points where the expression equals zero: \( x = -4 \), \( x = 0 \), \( x = 4 \).
4. **Solution in Interval Notation:**
The solution is \( [-4, 0] \cup [4, ∞) \).
Ensure each interval where the expression is non-negative is correctly identified and that edges of those intervals reflect equality conditions appropriately.
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