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

Answer :

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

We need to solve for [tex]\( x \)[/tex].

### Step 1: Isolate the Absolute Value

First, add 6 to both sides of the equation to isolate the absolute value expression:
[tex]\[
|x + 5| - 6 + 6 = 7 + 6
\][/tex]
[tex]\[
|x + 5| = 13
\][/tex]

### Step 2: Set Up Two Equations

The absolute value equation [tex]\( |x + 5| = 13 \)[/tex] means that [tex]\( x + 5 \)[/tex] could be 13 or [tex]\(-13\)[/tex]. So we set up two separate equations:

1. [tex]\( x + 5 = 13 \)[/tex]
2. [tex]\( x + 5 = -13 \)[/tex]

### Step 3: Solve Each Equation

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

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

### Step 4: Write Down the Solutions

The solutions to the equation [tex]\( |x + 5| - 6 = 7 \)[/tex] are:
[tex]\[
x = 8 \quad \text{and} \quad x = -18
\][/tex]

### Step 5: Match the Solutions to the Given Choices

Looking at the given options:

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

The correct answer is:
[tex]\[
\boxed{D}
\][/tex]