Answer :
Sure, let's solve the equation [tex]\(4|x + 6| + 8 = 28\)[/tex] step by step.
### Step 1: Isolate the Absolute Value Expression
First, let's subtract 8 from both sides of the equation to simplify it.
[tex]\[
4|x + 6| + 8 - 8 = 28 - 8
\][/tex]
[tex]\[
4|x + 6| = 20
\][/tex]
### Step 2: Solve for the Absolute Value
Next, we divide both sides by 4.
[tex]\[
\frac{4|x + 6|}{4} = \frac{20}{4}
\][/tex]
[tex]\[
|x + 6| = 5
\][/tex]
### Step 3: Set Up Two Equations
The absolute value [tex]\( |x + 6| = 5 \)[/tex] means that [tex]\( x + 6 \)[/tex] can be 5 or -5. Therefore, we set up two separate equations:
1. [tex]\( x + 6 = 5 \)[/tex]
2. [tex]\( x + 6 = -5 \)[/tex]
### Step 4: Solve Each Equation
For the first equation:
[tex]\[
x + 6 = 5
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 5 - 6
\][/tex]
[tex]\[
x = -1
\][/tex]
For the second equation:
[tex]\[
x + 6 = -5
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -5 - 6
\][/tex]
[tex]\[
x = -11
\][/tex]
### Step 5: List the Solutions
The solutions to the equation [tex]\(4|x + 6| + 8 = 28\)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].
So, the correct answer is:
A. [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex]
### Step 1: Isolate the Absolute Value Expression
First, let's subtract 8 from both sides of the equation to simplify it.
[tex]\[
4|x + 6| + 8 - 8 = 28 - 8
\][/tex]
[tex]\[
4|x + 6| = 20
\][/tex]
### Step 2: Solve for the Absolute Value
Next, we divide both sides by 4.
[tex]\[
\frac{4|x + 6|}{4} = \frac{20}{4}
\][/tex]
[tex]\[
|x + 6| = 5
\][/tex]
### Step 3: Set Up Two Equations
The absolute value [tex]\( |x + 6| = 5 \)[/tex] means that [tex]\( x + 6 \)[/tex] can be 5 or -5. Therefore, we set up two separate equations:
1. [tex]\( x + 6 = 5 \)[/tex]
2. [tex]\( x + 6 = -5 \)[/tex]
### Step 4: Solve Each Equation
For the first equation:
[tex]\[
x + 6 = 5
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = 5 - 6
\][/tex]
[tex]\[
x = -1
\][/tex]
For the second equation:
[tex]\[
x + 6 = -5
\][/tex]
Subtract 6 from both sides:
[tex]\[
x = -5 - 6
\][/tex]
[tex]\[
x = -11
\][/tex]
### Step 5: List the Solutions
The solutions to the equation [tex]\(4|x + 6| + 8 = 28\)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex].
So, the correct answer is:
A. [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex]