Question

Solve the absolute value inequality 2|x−8|−1<5 . Provide the answer using interval notation.(1 point)
The solution to 2|x−8|−1<5
is (

Answer 1

To solve the inequality \( 2|x - 8| - 1 < 5 \), we will follow the steps outlined below:

1. **Isolate the absolute value:**
\[
2|x - 8| - 1 < 5
\]
Add 1 to both sides:
\[
2|x - 8| < 6
\]
Divide both sides by 2:
\[
|x - 8| < 3
\]

2. **Remove the absolute value by writing two inequalities:**
\[
-3 < x - 8 < 3
\]

3. **Solve the inequalities:**
For the left part:
\[
-3 < x - 8 \quad \Rightarrow \quad x > 5
\]
For the right part:
\[
x - 8 < 3 \quad \Rightarrow \quad x < 11
\]

4. **Combine the results:**
\[
5 < x < 11
\]

5. **Write the solution in interval notation:**
The solution in interval notation is:
\[
(5, 11)
\]

So, the solution to the inequality \( 2|x - 8| - 1 < 5 \) is \((5, 11)\).