College

Solve [tex]|x+5|-6=7[/tex].

A. [tex]x=-8[/tex] and [tex]x=18[/tex]
B. [tex]x=8[/tex] and [tex]x=-18[/tex]
C. [tex]x=8[/tex] and [tex]x=-8[/tex]
D. [tex]x=-8[/tex] and [tex]x=-18[/tex]

Answer :

Let's solve the equation [tex]\( |x + 5| - 6 = 7 \)[/tex] step-by-step.

1. Isolate the absolute value expression:
[tex]\[
|x + 5| - 6 = 7
\][/tex]
Add 6 to both sides of the equation to isolate the absolute value:
[tex]\[
|x + 5| - 6 + 6 = 7 + 6
\][/tex]
[tex]\[
|x + 5| = 13
\][/tex]

2. Consider the definition of absolute value:
The absolute value equation [tex]\(|x + 5| = 13\)[/tex] can be split into two separate cases:
- Case 1: [tex]\( x + 5 = 13 \)[/tex]
- Case 2: [tex]\( x + 5 = -13 \)[/tex]

3. Solve for [tex]\(x\)[/tex] in each case:

Case 1: [tex]\( x + 5 = 13 \)[/tex]
[tex]\[
x + 5 = 13
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = 13 - 5
\][/tex]
[tex]\[
x = 8
\][/tex]

Case 2: [tex]\( x + 5 = -13 \)[/tex]
[tex]\[
x + 5 = -13
\][/tex]
Subtract 5 from both sides:
[tex]\[
x = -13 - 5
\][/tex]
[tex]\[
x = -18
\][/tex]

4. Write the solutions:
The solutions to the equation [tex]\( |x + 5| - 6 = 7 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = -18 \)[/tex].

Therefore, the correct answer is:
[tex]\[
\boxed{x = 8 \text{ and } x = -18}
\][/tex]

The correct choice from the given options is:
B. [tex]\( x = 8 \)[/tex] and [tex]\( x = -18 \)[/tex]