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Solve [tex]4|x+5| + 8 = 24[/tex].

A. [tex]x = 1[/tex] and [tex]x = -1[/tex]
B. [tex]x = 1[/tex] and [tex]x = ?[/tex]
C. [tex]x = -1[/tex] and [tex]x = 9[/tex]
D. [tex]x = -1[/tex] and [tex]x = 9[/tex]

Answer :

Sure, let's tackle the equation step by step.

Equation: [tex]\(4|x+5| + 8 = 24\)[/tex]

### Step 1: Isolate the absolute value term.

Subtract 8 from both sides of the equation:

[tex]\[ 4|x+5| + 8 - 8 = 24 - 8 \][/tex]

Which simplifies to:

[tex]\[ 4|x+5| = 16 \][/tex]

### Step 2: Solve for [tex]\( |x+5| \)[/tex].

Divide both sides of the equation by 4:

[tex]\[ \frac{4|x+5|}{4} = \frac{16}{4} \][/tex]

This simplifies to:

[tex]\[ |x+5| = 4 \][/tex]

### Step 3: Break it down into two separate linear equations.

The absolute value equation [tex]\( |x+5| = 4 \)[/tex] can be split into two equations:

1. [tex]\( x + 5 = 4 \)[/tex]
2. [tex]\( x + 5 = -4 \)[/tex]

### Step 4: Solve each equation separately.

For the first equation [tex]\( x + 5 = 4 \)[/tex]:

[tex]\[ x = 4 - 5 \][/tex]
[tex]\[ x = -1 \][/tex]

For the second equation [tex]\( x + 5 = -4 \)[/tex]:

[tex]\[ x = -4 - 5 \][/tex]
[tex]\[ x = -9 \][/tex]

### Step 5: Conclusion

The solutions to [tex]\( 4|x+5| + 8 = 24 \)[/tex] are:

[tex]\[ x = -1 \text{ and } x = -9 \][/tex]

### Step 6: Match the solutions with the given choices.

- A. [tex]\( x = 1 \)[/tex] and [tex]\( x = -1 \)[/tex] (not correct)
- B. [tex]\( x = 1 \)[/tex] and [tex]\( x = \, ? \)[/tex] (incomplete)
- C. [tex]\( x = -1 \)[/tex] and [tex]\( x = 9 \)[/tex] (not correct)
- D. [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex] (correct)

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

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