Answer :
To solve the equation [tex]\(4|x+5|=16\)[/tex], let's break it down step-by-step:
1. Isolate the absolute value:
Start by dividing both sides of the equation by 4 to simplify. This gives:
[tex]\[
|x+5| = 4
\][/tex]
2. Consider the two cases for the absolute value:
The expression [tex]\(|x+5| = 4\)[/tex] leads to two separate equations because absolute value represents the distance from zero, accounting for both the positive and negative direction.
- Case 1: [tex]\(x+5 = 4\)[/tex]
- Solve for [tex]\(x\)[/tex] by subtracting 5 from both sides:
[tex]\[
x = 4 - 5
\][/tex]
Simplifying gives:
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\(x+5 = -4\)[/tex]
- Similarly, solve for [tex]\(x\)[/tex] by subtracting 5 from both sides:
[tex]\[
x = -4 - 5
\][/tex]
Simplifying gives:
[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].
Therefore, the correct answer is:
C. [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 simplify. This gives:
[tex]\[
|x+5| = 4
\][/tex]
2. Consider the two cases for the absolute value:
The expression [tex]\(|x+5| = 4\)[/tex] leads to two separate equations because absolute value represents the distance from zero, accounting for both the positive and negative direction.
- Case 1: [tex]\(x+5 = 4\)[/tex]
- Solve for [tex]\(x\)[/tex] by subtracting 5 from both sides:
[tex]\[
x = 4 - 5
\][/tex]
Simplifying gives:
[tex]\[
x = -1
\][/tex]
- Case 2: [tex]\(x+5 = -4\)[/tex]
- Similarly, solve for [tex]\(x\)[/tex] by subtracting 5 from both sides:
[tex]\[
x = -4 - 5
\][/tex]
Simplifying gives:
[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].
Therefore, the correct answer is:
C. [tex]\(x = -1\)[/tex] and [tex]\(x = -9\)[/tex]