Answer :
Sure, let's solve the equation step-by-step:
The given equation is:
[tex]\[ 4|x + 7| + 8 = 32 \][/tex]
1. Isolate the absolute value term:
First, subtract 8 from both sides of the equation:
[tex]\[ 4|x + 7| + 8 - 8 = 32 - 8 \][/tex]
[tex]\[ 4|x + 7| = 24 \][/tex]
2. Divide both sides by 4 to simplify:
[tex]\[ \frac{4|x + 7|}{4} = \frac{24}{4} \][/tex]
[tex]\[ |x + 7| = 6 \][/tex]
3. Solve the absolute value equation:
The absolute value equation [tex]\( |x + 7| = 6 \)[/tex] means that:
[tex]\[ x + 7 = 6 \quad \text{or} \quad x + 7 = -6 \][/tex]
4. Solve each equation separately:
- For [tex]\( x + 7 = 6 \)[/tex]:
[tex]\[ x = 6 - 7 \][/tex]
[tex]\[ x = -1 \][/tex]
- For [tex]\( x + 7 = -6 \)[/tex]:
[tex]\[ x = -6 - 7 \][/tex]
[tex]\[ x = -13 \][/tex]
So, the solutions to the equation [tex]\( 4|x + 7| + 8 = 32 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -13 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{D. \ x = -1 \ \text{and} \ x = -13} \][/tex]
The given equation is:
[tex]\[ 4|x + 7| + 8 = 32 \][/tex]
1. Isolate the absolute value term:
First, subtract 8 from both sides of the equation:
[tex]\[ 4|x + 7| + 8 - 8 = 32 - 8 \][/tex]
[tex]\[ 4|x + 7| = 24 \][/tex]
2. Divide both sides by 4 to simplify:
[tex]\[ \frac{4|x + 7|}{4} = \frac{24}{4} \][/tex]
[tex]\[ |x + 7| = 6 \][/tex]
3. Solve the absolute value equation:
The absolute value equation [tex]\( |x + 7| = 6 \)[/tex] means that:
[tex]\[ x + 7 = 6 \quad \text{or} \quad x + 7 = -6 \][/tex]
4. Solve each equation separately:
- For [tex]\( x + 7 = 6 \)[/tex]:
[tex]\[ x = 6 - 7 \][/tex]
[tex]\[ x = -1 \][/tex]
- For [tex]\( x + 7 = -6 \)[/tex]:
[tex]\[ x = -6 - 7 \][/tex]
[tex]\[ x = -13 \][/tex]
So, the solutions to the equation [tex]\( 4|x + 7| + 8 = 32 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -13 \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{D. \ x = -1 \ \text{and} \ x = -13} \][/tex]