High School

Solve [tex]$4|x+6|+8=28$[/tex]

A. [tex]$x=1$[/tex] and [tex][tex]$x=-1$[/tex][/tex]
B. [tex]$x=1$[/tex] and [tex]$x=-11$[/tex]
C. [tex][tex]$x=-1$[/tex][/tex] and [tex]$x=-11$[/tex]
D. [tex]$x=-1$[/tex] and [tex][tex]$x=11$[/tex][/tex]

Answer :

Let's solve the equation step-by-step:

The given equation is:
[tex]\[ 4|x+6| + 8 = 28 \][/tex]

### Step 1: Isolate the absolute value term
First, subtract 8 from both sides of the equation:
[tex]\[ 4|x+6| + 8 - 8 = 28 - 8 \][/tex]
[tex]\[ 4|x+6| = 20 \][/tex]

### Step 2: Solve for the absolute value
Now, divide both sides of the equation by 4:
[tex]\[ |x+6| = \frac{20}{4} \][/tex]
[tex]\[ |x+6| = 5 \][/tex]

### Step 3: Set up two separate equations
Since the absolute value of [tex]\( x+6 \)[/tex] is 5, we can set up two equations:
[tex]\[ x + 6 = 5 \][/tex]
[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: Write the solutions
So, the solutions to the equation [tex]\( 4|x+6| + 8 = 28 \)[/tex] are:
[tex]\[ x = -1 \][/tex] and [tex]\[ x = -11 \][/tex]

The correct answer is:
C. [tex]\( x = -1 \)[/tex] and [tex]\( x = -11 \)[/tex]