Answer :
To solve the equation [tex]\(4|x+5|=16\)[/tex], let's break it down into simpler steps:
1. Isolate the Absolute Value: Start by dividing both sides of the equation by 4 to isolate the absolute value expression:
[tex]\[
|x+5| = \frac{16}{4} = 4
\][/tex]
2. Set Up the Two Cases: An absolute value equation like [tex]\( |x+5| = 4 \)[/tex] splits into two possible equations. This is because the expression inside the absolute value can be either positive or negative.
- Case 1: When [tex]\( x+5 = 4 \)[/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x+5 = 4 \\
x = 4 - 5 \\
x = -1
\][/tex]
- Case 2: When [tex]\( x+5 = -4 \)[/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x+5 = -4 \\
x = -4 - 5 \\
x = -9
\][/tex]
3. Conclusion: The solutions to the equation are:
[tex]\[
x = -1 \quad \text{and} \quad x = -9
\][/tex]
Thus, the correct answer is option D: [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
1. Isolate the Absolute Value: Start by dividing both sides of the equation by 4 to isolate the absolute value expression:
[tex]\[
|x+5| = \frac{16}{4} = 4
\][/tex]
2. Set Up the Two Cases: An absolute value equation like [tex]\( |x+5| = 4 \)[/tex] splits into two possible equations. This is because the expression inside the absolute value can be either positive or negative.
- Case 1: When [tex]\( x+5 = 4 \)[/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x+5 = 4 \\
x = 4 - 5 \\
x = -1
\][/tex]
- Case 2: When [tex]\( x+5 = -4 \)[/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
x+5 = -4 \\
x = -4 - 5 \\
x = -9
\][/tex]
3. Conclusion: The solutions to the equation are:
[tex]\[
x = -1 \quad \text{and} \quad x = -9
\][/tex]
Thus, the correct answer is option D: [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].