Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Solving the Compound Inequality
Solve the following inequality:
\[ 5|4x - 2| + 4 \leq 9 \]
**Explanation:**
1. **Start by isolating the absolute value expression:**
\[ 5|4x - 2| + 4 \leq 9 \]
Subtract 4 from both sides to isolate the absolute value term:
\[ 5|4x - 2| \leq 5 \]
2. **Divide both sides by 5:**
\[ |4x - 2| \leq 1 \]
3. **Solve the absolute value inequality:**
Given the absolute value inequality \( |A| \leq B \), this translates to:
\[ -B \leq A \leq B \]
For our inequality \( |4x - 2| \leq 1 \), this translates to:
\[ -1 \leq 4x - 2 \leq 1 \]
4. **Solve the compound inequality:**
First inequality:
\[ -1 \leq 4x - 2 \]
Add 2 to both sides:
\[ 1 \leq 4x \]
Divide by 4:
\[ \frac{1}{4} \leq x \]
Second inequality:
\[ 4x - 2 \leq 1 \]
Add 2 to both sides:
\[ 4x \leq 3 \]
Divide by 4:
\[ x \leq \frac{3}{4} \]
5. **Combine the solutions:**
\[ \frac{1}{4} \leq x \leq \frac{3}{4} \]
Thus, the solution to the inequality \( 5|4x - 2| + 4 \leq 9 \) is:
\[ \frac{1}{4} \leq x \leq \frac{3}{4} \]
This interval notation denotes that \( x \) is between \(\frac{1}{4}\) and \(\frac{3}{4}\) inclusive.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6bf0e7cf-f0aa-4881-86db-f1b5f420fddf%2F9e4d5650-7ed4-466d-8977-aa7a5c142354%2Fi0jgyxw.jpeg&w=3840&q=75)
Transcribed Image Text:### Solving the Compound Inequality
Solve the following inequality:
\[ 5|4x - 2| + 4 \leq 9 \]
**Explanation:**
1. **Start by isolating the absolute value expression:**
\[ 5|4x - 2| + 4 \leq 9 \]
Subtract 4 from both sides to isolate the absolute value term:
\[ 5|4x - 2| \leq 5 \]
2. **Divide both sides by 5:**
\[ |4x - 2| \leq 1 \]
3. **Solve the absolute value inequality:**
Given the absolute value inequality \( |A| \leq B \), this translates to:
\[ -B \leq A \leq B \]
For our inequality \( |4x - 2| \leq 1 \), this translates to:
\[ -1 \leq 4x - 2 \leq 1 \]
4. **Solve the compound inequality:**
First inequality:
\[ -1 \leq 4x - 2 \]
Add 2 to both sides:
\[ 1 \leq 4x \]
Divide by 4:
\[ \frac{1}{4} \leq x \]
Second inequality:
\[ 4x - 2 \leq 1 \]
Add 2 to both sides:
\[ 4x \leq 3 \]
Divide by 4:
\[ x \leq \frac{3}{4} \]
5. **Combine the solutions:**
\[ \frac{1}{4} \leq x \leq \frac{3}{4} \]
Thus, the solution to the inequality \( 5|4x - 2| + 4 \leq 9 \) is:
\[ \frac{1}{4} \leq x \leq \frac{3}{4} \]
This interval notation denotes that \( x \) is between \(\frac{1}{4}\) and \(\frac{3}{4}\) inclusive.
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