Answer :
We start with the equation
[tex]$$
|x - 5| + 7 = 15.
$$[/tex]
Step 1. Isolate the absolute value expression
Subtract 7 from both sides to get
[tex]$$
|x - 5| = 15 - 7 = 8.
$$[/tex]
Step 2. Solve the absolute value equation
The definition of absolute value gives us two cases:
1. When the expression inside is positive:
[tex]$$
x - 5 = 8.
$$[/tex]
Solving for [tex]$x$[/tex], add 5 to both sides:
[tex]$$
x = 8 + 5 = 13.
$$[/tex]
2. When the expression inside is negative:
[tex]$$
x - 5 = -8.
$$[/tex]
Solving for [tex]$x$[/tex], add 5 to both sides:
[tex]$$
x = -8 + 5 = -3.
$$[/tex]
Step 3. Conclude the solutions
The two solutions for the equation are
[tex]$$
x = 13 \quad \text{and} \quad x = -3.
$$[/tex]
Thus, the correct answer is:
A. [tex]$x=13$[/tex] and [tex]$x=-3$[/tex].
[tex]$$
|x - 5| + 7 = 15.
$$[/tex]
Step 1. Isolate the absolute value expression
Subtract 7 from both sides to get
[tex]$$
|x - 5| = 15 - 7 = 8.
$$[/tex]
Step 2. Solve the absolute value equation
The definition of absolute value gives us two cases:
1. When the expression inside is positive:
[tex]$$
x - 5 = 8.
$$[/tex]
Solving for [tex]$x$[/tex], add 5 to both sides:
[tex]$$
x = 8 + 5 = 13.
$$[/tex]
2. When the expression inside is negative:
[tex]$$
x - 5 = -8.
$$[/tex]
Solving for [tex]$x$[/tex], add 5 to both sides:
[tex]$$
x = -8 + 5 = -3.
$$[/tex]
Step 3. Conclude the solutions
The two solutions for the equation are
[tex]$$
x = 13 \quad \text{and} \quad x = -3.
$$[/tex]
Thus, the correct answer is:
A. [tex]$x=13$[/tex] and [tex]$x=-3$[/tex].