To solve the equation [tex]\( |x - 5| + 7 = 13 \)[/tex], we will first isolate the absolute value term and then consider the two possible cases for the absolute value expression.
1. Start by isolating the absolute value term:
[tex]\[
|x - 5| + 7 = 13
\][/tex]
Subtract 7 from both sides:
[tex]\[
|x - 5| = 6
\][/tex]
2. Now, we need to solve the equation [tex]\( |x - 5| = 6 \)[/tex]. The absolute value equation [tex]\( |A| = B \)[/tex] implies two cases:
- [tex]\( A = B \)[/tex]
- [tex]\( A = -B \)[/tex]
Applying this to our equation [tex]\( |x - 5| = 6 \)[/tex], we get:
- [tex]\( x - 5 = 6 \)[/tex]
- [tex]\( x - 5 = -6 \)[/tex]
3. Solve each case separately:
Case 1: [tex]\( x - 5 = 6 \)[/tex]
[tex]\[
x - 5 = 6
\][/tex]
Add 5 to both sides:
[tex]\[
x = 11
\][/tex]
Case 2: [tex]\( x - 5 = -6 \)[/tex]
[tex]\[
x - 5 = -6
\][/tex]
Add 5 to both sides:
[tex]\[
x = -1
\][/tex]
4. Hence, the solutions to the equation [tex]\( |x - 5| + 7 = 13 \)[/tex] are [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex].
Therefore, the correct answer is:
A. [tex]\( x = 11 \)[/tex] and [tex]\( x = -1 \)[/tex]
