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Solve [tex]$4|x+5|=24$[/tex]

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

Answer :

Sure! Let's solve the equation [tex]\(4|x + 5| = 24\)[/tex] step-by-step.

### Step 1: Isolate the absolute value expression.
First, we need to get the absolute value expression by itself:
[tex]\[ 4|x + 5| = 24 \][/tex]

Divide both sides by 4:
[tex]\[ |x + 5| = \frac{24}{4} \][/tex]
[tex]\[ |x + 5| = 6 \][/tex]

### Step 2: Split into two separate equations.
The expression inside the absolute value can be either positive or negative:
[tex]\[ x + 5 = 6 \][/tex]
[tex]\[ x + 5 = -6 \][/tex]

### Step 3: Solve each equation separately.

#### Equation 1:
[tex]\[ x + 5 = 6 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = 6 - 5 \][/tex]
[tex]\[ x = 1 \][/tex]

#### Equation 2:
[tex]\[ x + 5 = -6 \][/tex]
Subtract 5 from both sides:
[tex]\[ x = -6 - 5 \][/tex]
[tex]\[ x = -11 \][/tex]

### Step 4: Combine the solutions.
The solutions to the equation [tex]\(4|x + 5| = 24\)[/tex] are:
[tex]\[ x = 1 \text{ and } x = -11 \][/tex]

### Verify the solutions with the options given:
A. [tex]\(x = 11\)[/tex] and [tex]\(x = -11\)[/tex] - Incorrect.
B. [tex]\(x = -11\)[/tex] and [tex]\(x = -1\)[/tex] - Incorrect.
C. [tex]\(x = 11\)[/tex] and [tex]\(x = -1\)[/tex] - Incorrect.
D. [tex]\(x = -11\)[/tex] and [tex]\(x = 1\)[/tex] - Correct.

So, the correct answer is:
[tex]\[ \boxed{D} \][/tex]

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