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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4.)
b
![b. \( |4x - 5| - 3 > 10 \)
This expression represents an absolute value inequality. The goal is to determine the values of \( x \) that satisfy the inequality. To solve this, isolate the absolute value on one side and then consider the two possible cases for the expression inside the absolute value.
1. **Isolate the Absolute Value:**
\[ |4x - 5| > 13 \]
2. **Consider the Two Cases:**
- Case 1: \( 4x - 5 > 13 \)
- Case 2: \( 4x - 5 < -13 \)
3. **Solve Each Case:**
- **Case 1:**
\[ 4x - 5 > 13 \]
\[ 4x > 18 \]
\[ x > \frac{18}{4} \]
\[ x > 4.5 \]
- **Case 2:**
\[ 4x - 5 < -13 \]
\[ 4x < -8 \]
\[ x < \frac{-8}{4} \]
\[ x < -2 \]
4. **Combine the Solutions:**
The solution to the inequality is \( x < -2 \) or \( x > 4.5 \).
This showcases how to handle absolute value inequalities and find the solution for the variable \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5f7b3938-790b-4169-a858-04369df79473%2Ff02947be-dab2-4dd4-99dc-d658a6f2886a%2Fbha5g79_processed.png&w=3840&q=75)
Transcribed Image Text:b. \( |4x - 5| - 3 > 10 \)
This expression represents an absolute value inequality. The goal is to determine the values of \( x \) that satisfy the inequality. To solve this, isolate the absolute value on one side and then consider the two possible cases for the expression inside the absolute value.
1. **Isolate the Absolute Value:**
\[ |4x - 5| > 13 \]
2. **Consider the Two Cases:**
- Case 1: \( 4x - 5 > 13 \)
- Case 2: \( 4x - 5 < -13 \)
3. **Solve Each Case:**
- **Case 1:**
\[ 4x - 5 > 13 \]
\[ 4x > 18 \]
\[ x > \frac{18}{4} \]
\[ x > 4.5 \]
- **Case 2:**
\[ 4x - 5 < -13 \]
\[ 4x < -8 \]
\[ x < \frac{-8}{4} \]
\[ x < -2 \]
4. **Combine the Solutions:**
The solution to the inequality is \( x < -2 \) or \( x > 4.5 \).
This showcases how to handle absolute value inequalities and find the solution for the variable \( x \).
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