Answer :
Sure! Let's solve the equation [tex]\(4|x+5| = 16\)[/tex] step by step.
1. Divide both sides by 4 to isolate the absolute value:
[tex]\[
\frac{4|x+5|}{4} = \frac{16}{4}
\][/tex]
[tex]\[
|x+5| = 4
\][/tex]
2. Consider the definition of absolute value, which means that [tex]\(x+5\)[/tex] could be either 4 or -4. This gives us two separate equations to solve:
Case 1: [tex]\(x + 5 = 4\)[/tex]
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
Case 2: [tex]\(x + 5 = -4\)[/tex]
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
So, the solutions to the equation [tex]\(4|x+5| = 16\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
The correct answer is:
[tex]\[
\boxed{x = -1 \text{ and } x = -9}
\][/tex]
This matches with option B from the provided choices:
B. [tex]\(\boxed{x = -1 \text{ and } x = -9}\)[/tex]
1. Divide both sides by 4 to isolate the absolute value:
[tex]\[
\frac{4|x+5|}{4} = \frac{16}{4}
\][/tex]
[tex]\[
|x+5| = 4
\][/tex]
2. Consider the definition of absolute value, which means that [tex]\(x+5\)[/tex] could be either 4 or -4. This gives us two separate equations to solve:
Case 1: [tex]\(x + 5 = 4\)[/tex]
[tex]\[
x + 5 = 4
\][/tex]
Subtract 5 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 4 - 5
\][/tex]
[tex]\[
x = -1
\][/tex]
Case 2: [tex]\(x + 5 = -4\)[/tex]
[tex]\[
x + 5 = -4
\][/tex]
Subtract 5 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -4 - 5
\][/tex]
[tex]\[
x = -9
\][/tex]
So, the solutions to the equation [tex]\(4|x+5| = 16\)[/tex] are [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex].
The correct answer is:
[tex]\[
\boxed{x = -1 \text{ and } x = -9}
\][/tex]
This matches with option B from the provided choices:
B. [tex]\(\boxed{x = -1 \text{ and } x = -9}\)[/tex]