To solve the equation [tex]\( |x - 5| + 7 = 17 \)[/tex], follow these steps:
1. Isolate the absolute value expression:
Start with the given equation:
[tex]\[
|x - 5| + 7 = 17
\][/tex]
Subtract 7 from both sides to isolate the absolute value term:
[tex]\[
|x - 5| = 17 - 7
\][/tex]
Simplifying this gives:
[tex]\[
|x - 5| = 10
\][/tex]
2. Break down the absolute value equation into two cases:
The absolute value equation [tex]\( |x - 5| = 10 \)[/tex] implies two scenarios:
- Case 1: [tex]\( x - 5 = 10 \)[/tex]
- Case 2: [tex]\( x - 5 = -10 \)[/tex]
3. Solve each case separately:
- For Case 1:
[tex]\[
x - 5 = 10
\][/tex]
Add 5 to both sides:
[tex]\[
x = 10 + 5
\][/tex]
[tex]\[
x = 15
\][/tex]
- For Case 2:
[tex]\[
x - 5 = -10
\][/tex]
Add 5 to both sides:
[tex]\[
x = -10 + 5
\][/tex]
[tex]\[
x = -5
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\( |x - 5| + 7 = 17 \)[/tex] are [tex]\( x = 15 \)[/tex] and [tex]\( x = -5 \)[/tex].
Therefore, the correct option is:
[tex]\[
\boxed{C. \, x=15 \text{ and } x=-5}
\][/tex]
