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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Question
![**Problem:**
Solve the inequality:
\[ 5 - 2|3x - 5| > -1 \]
**Solution:**
To solve this inequality, follow these steps:
1. **Isolate the Absolute Value:**
Start by isolating the absolute value expression. Add 1 to both sides:
\[ 5 - 2|3x - 5| + 1 > 0 \]
Simplify:
\[ 6 - 2|3x - 5| > 0 \]
2. **Rearrange Terms:**
Subtract 6 from both sides:
\[ -2|3x - 5| > -6 \]
3. **Divide by -2:**
Divide each side by -2, remembering to reverse the inequality sign:
\[ |3x - 5| < 3 \]
4. **Solve the Absolute Value Inequality:**
The inequality \( |A| < B \) implies \( -B < A < B \). Therefore:
\[ -3 < 3x - 5 < 3 \]
5. **Solve for \( x \):**
Split the inequality into two parts and solve each:
- \( -3 < 3x - 5 \):
Add 5 to both sides:
\[ 2 < 3x \]
Divide by 3:
\[ \frac{2}{3} < x \]
- \( 3x - 5 < 3 \):
Add 5 to both sides:
\[ 3x < 8 \]
Divide by 3:
\[ x < \frac{8}{3} \]
6. **Combine the Results:**
The solution to the inequality is:
\[ \frac{2}{3} < x < \frac{8}{3} \]
**Conclusion:**
This means that the variable \( x \) must be greater than \(\frac{2}{3}\) and less than \(\frac{8}{3}\) for the original inequality to hold. Thus, the interval \(\left( \frac{2}{3}, \frac{8}{3} \right)\) represents the solution set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2a49397d-2847-420e-bdfe-93d5dc178c48%2Fcccd96df-a013-4bdb-a7b8-12ff469928ba%2F0twfxho_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Solve the inequality:
\[ 5 - 2|3x - 5| > -1 \]
**Solution:**
To solve this inequality, follow these steps:
1. **Isolate the Absolute Value:**
Start by isolating the absolute value expression. Add 1 to both sides:
\[ 5 - 2|3x - 5| + 1 > 0 \]
Simplify:
\[ 6 - 2|3x - 5| > 0 \]
2. **Rearrange Terms:**
Subtract 6 from both sides:
\[ -2|3x - 5| > -6 \]
3. **Divide by -2:**
Divide each side by -2, remembering to reverse the inequality sign:
\[ |3x - 5| < 3 \]
4. **Solve the Absolute Value Inequality:**
The inequality \( |A| < B \) implies \( -B < A < B \). Therefore:
\[ -3 < 3x - 5 < 3 \]
5. **Solve for \( x \):**
Split the inequality into two parts and solve each:
- \( -3 < 3x - 5 \):
Add 5 to both sides:
\[ 2 < 3x \]
Divide by 3:
\[ \frac{2}{3} < x \]
- \( 3x - 5 < 3 \):
Add 5 to both sides:
\[ 3x < 8 \]
Divide by 3:
\[ x < \frac{8}{3} \]
6. **Combine the Results:**
The solution to the inequality is:
\[ \frac{2}{3} < x < \frac{8}{3} \]
**Conclusion:**
This means that the variable \( x \) must be greater than \(\frac{2}{3}\) and less than \(\frac{8}{3}\) for the original inequality to hold. Thus, the interval \(\left( \frac{2}{3}, \frac{8}{3} \right)\) represents the solution set.
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