Let's solve the given absolute value inequality step by step:
Given:
[tex]\[
\frac{|x+15|}{5} \leq 26
\][/tex]
### Step 1: Clear the fraction
Multiply both sides of the inequality by 5 to eliminate the fraction:
[tex]\[
|x + 15| \leq 130
\][/tex]
### Step 2: Break down the absolute value inequality
The inequality [tex]\( |x + 15| \leq 130 \)[/tex] means that the expression inside the absolute value, [tex]\(x + 15\)[/tex], lies between [tex]\(-130\)[/tex] and [tex]\(130\)[/tex]. This can be written as two separate inequalities:
[tex]\[
-130 \leq x + 15 \leq 130
\][/tex]
### Step 3: Solve each inequality separately
First, solve the inequality on the left:
[tex]\[
-130 \leq x + 15
\][/tex]
Subtract 15 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
-130 - 15 \leq x
\][/tex]
[tex]\[
-145 \leq x
\][/tex]
Second, solve the inequality on the right:
[tex]\[
x + 15 \leq 130
\][/tex]
Subtract 15 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x \leq 130 - 15
\][/tex]
[tex]\[
x \leq 115
\][/tex]
### Conclusion
Combining the two inequalities, we have:
[tex]\[
-145 \leq x \leq 115
\][/tex]
### Positive Absolute Value
We need the positive absolute value for [tex]\(x\)[/tex]. Hence, we look at the upper bound of this range:
[tex]\[
x \leq 115
\][/tex]
Thus, the positive absolute value for [tex]\(x\)[/tex] is:
[tex]\[
115
\][/tex]
