To solve the equation [tex]\(|x - 5| = 7\)[/tex], we need to consider that the absolute value expression has two possible scenarios. The absolute value equation [tex]\(|A| = B\)[/tex] translates into two equations: [tex]\(A = B\)[/tex] and [tex]\(A = -B\)[/tex].
Let's apply this principle to our given equation:
[tex]\[
|x - 5| = 7
\][/tex]
This can be broken down into two separate equations:
1. [tex]\(x - 5 = 7\)[/tex]
2. [tex]\(x - 5 = -7\)[/tex]
We will solve each equation individually:
### Solving the first equation:
[tex]\[
x - 5 = 7
\][/tex]
Add 5 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 7 + 5
\][/tex]
[tex]\[
x = 12
\][/tex]
### Solving the second equation:
[tex]\[
x - 5 = -7
\][/tex]
Add 5 to both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = -7 + 5
\][/tex]
[tex]\[
x = -2
\][/tex]
Thus, the solutions to the equation [tex]\(|x - 5| = 7\)[/tex] are [tex]\(x = 12\)[/tex] and [tex]\(x = -2\)[/tex].
The correct answer is:
D. [tex]\(x = 12\)[/tex] and [tex]\(x = -2\)[/tex]
