Answer :
To solve the equation [tex]\(4|x+5| = 16\)[/tex], let's break it down step-by-step:
1. Isolate the absolute value:
Divide both sides of the equation by 4:
[tex]\[
|x+5| = \frac{16}{4} = 4
\][/tex]
2. Set up two separate equations:
The expression [tex]\( |x+5| = 4 \)[/tex] means that [tex]\( x+5 \)[/tex] can be either 4 or -4, because absolute value represents the distance from zero and can be positive or negative.
- Case 1: [tex]\( x+5 = 4 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\( x+5 = -4 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
3. Conclusion:
The solutions to the equation [tex]\( 4|x+5| = 16 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].
From the options provided, the correct answer is C. [tex]\(x=-1\)[/tex] and [tex]\(x=-9\)[/tex].
1. Isolate the absolute value:
Divide both sides of the equation by 4:
[tex]\[
|x+5| = \frac{16}{4} = 4
\][/tex]
2. Set up two separate equations:
The expression [tex]\( |x+5| = 4 \)[/tex] means that [tex]\( x+5 \)[/tex] can be either 4 or -4, because absolute value represents the distance from zero and can be positive or negative.
- Case 1: [tex]\( x+5 = 4 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\( x+5 = -4 \)[/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
3. Conclusion:
The solutions to the equation [tex]\( 4|x+5| = 16 \)[/tex] are [tex]\( x = -1 \)[/tex] and [tex]\( x = -9 \)[/tex].
From the options provided, the correct answer is C. [tex]\(x=-1\)[/tex] and [tex]\(x=-9\)[/tex].