To solve the equation [tex]\( |x-5| + 7 = 15 \)[/tex], let's break the problem into a few manageable steps.
### Step 1: Isolate the Absolute Value Expression
First, we need to isolate the absolute value expression [tex]\( |x - 5| \)[/tex]. To do this, we'll subtract 7 from both sides of the equation:
[tex]\[
|x - 5| + 7 - 7 = 15 - 7
\][/tex]
This simplifies to:
[tex]\[
|x - 5| = 8
\][/tex]
### Step 2: Solve the Absolute Value Equation
The absolute value equation [tex]\( |x - 5| = 8 \)[/tex] means that the expression inside the absolute value can be either [tex]\( 8 \)[/tex] or [tex]\( -8 \)[/tex]. So, we set up two separate equations:
[tex]\[
x - 5 = 8 \quad \text{or} \quad x - 5 = -8
\][/tex]
#### Solving the First Equation
[tex]\[
x - 5 = 8
\][/tex]
Add 5 to both sides:
[tex]\[
x = 8 + 5
\][/tex]
[tex]\[
x = 13
\][/tex]
#### Solving the Second Equation
[tex]\[
x - 5 = -8
\][/tex]
Add 5 to both sides:
[tex]\[
x = -8 + 5
\][/tex]
[tex]\[
x = -3
\][/tex]
### Step 3: Verify the Solutions
The potential solutions are [tex]\( x = 13 \)[/tex] and [tex]\( x = -3 \)[/tex]. To ensure these are correct, we substitute them back into the original equation:
#### For [tex]\( x = 13 \)[/tex]
[tex]\[
|13 - 5| + 7 = |8| + 7 = 8 + 7 = 15
\][/tex]
This holds true.
#### For [tex]\( x = -3 \)[/tex]
[tex]\[
|-3 - 5| + 7 = |-8| + 7 = 8 + 7 = 15
\][/tex]
This also holds true.
### Step 4: Conclusion
Both solutions satisfy the original equation. Therefore, the correct answers are:
[tex]\[
x = 13 \quad \text{and} \quad x = -3
\][/tex]
Looking at the given answer choices, the correct answer is:
[tex]\[
\boxed{D}
\][/tex]
