
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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.](https://content.bartleby.com/qna-images/question/6bf0e7cf-f0aa-4881-86db-f1b5f420fddf/9e4d5650-7ed4-466d-8977-aa7a5c142354/i0jgyxw_thumbnail.jpeg)
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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