Question
(8) Solve the inequality: \( |2 w-5|+4>13 \)
Ask by Vargas Burgess. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solution to the inequality \( |2w - 5| + 4 > 13 \) is all real numbers \( w \) such that \( w < -2 \) or \( w > 7 \).
Solution
To solve the inequality \( |2w - 5| + 4 > 13 \), follow these steps:
1. **Isolate the Absolute Value:**
\[
|2w - 5| + 4 > 13
\]
Subtract 4 from both sides:
\[
|2w - 5| > 9
\]
2. **Break Down the Absolute Value Inequality:**
The inequality \( |A| > B \) (where \( B > 0 \)) can be rewritten as two separate inequalities:
\[
A > B \quad \text{or} \quad A < -B
\]
Applying this to our problem:
\[
2w - 5 > 9 \quad \text{or} \quad 2w - 5 < -9
\]
3. **Solve Each Inequality Separately:**
- **First Inequality:**
\[
2w - 5 > 9
\]
Add 5 to both sides:
\[
2w > 14
\]
Divide by 2:
\[
w > 7
\]
- **Second Inequality:**
\[
2w - 5 < -9
\]
Add 5 to both sides:
\[
2w < -4
\]
Divide by 2:
\[
w < -2
\]
4. **Combine the Solutions:**
The solution to the inequality is:
\[
w < -2 \quad \text{or} \quad w > 7
\]
**Final Answer:**
All real numbers w with w < –2 or w greater than seven. In other words, w < −2 or w > 7
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To solve the inequality \( |2w - 5| + 4 > 13 \), we first isolate the absolute value by subtracting 4 from both sides: \[ |2w - 5| > 9. \] Next, we need to consider the two cases for the absolute value: 1. \( 2w - 5 > 9 \) 2. \( 2w - 5 < -9 \) For the first case: \[ 2w - 5 > 9 \] Adding 5 to both sides gives: \[ 2w > 14. \] Dividing both sides by 2 yields: \[ w > 7. \] For the second case: \[ 2w - 5 < -9 \] Adding 5 to both sides gives: \[ 2w < -4. \] Dividing both sides by 2 yields: \[ w < -2. \] Thus, the solution to the inequality \( |2w - 5| + 4 > 13 \) is: \[ w < -2 \quad \text{or} \quad w > 7. \]
