Answer :
To solve the equation [tex]\(5|x+9|=80\)[/tex], we need to consider the nature of absolute value, which can have two scenarios: when the expression inside is positive or when it is negative.
1. Positive Case:
- Assume [tex]\(x + 9\)[/tex] is positive or zero. The equation becomes:
[tex]\[
5(x + 9) = 80
\][/tex]
- Divide both sides by 5 to simplify:
[tex]\[
x + 9 = 16
\][/tex]
- Subtract 9 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 16 - 9 = 7
\][/tex]
2. Negative Case:
- Assume [tex]\(x + 9\)[/tex] is negative. Thus, we consider:
[tex]\[
5(-(x + 9)) = 80
\][/tex]
Which simplifies to:
[tex]\[
-5(x + 9) = 80
\][/tex]
- Divide both sides by -5:
[tex]\[
x + 9 = -16
\][/tex]
- Subtract 9 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -16 - 9 = -25
\][/tex]
So, the solutions to the equation [tex]\(5|x+9|=80\)[/tex] are [tex]\(x = 7\)[/tex] or [tex]\(x = -25\)[/tex].
Therefore, the correct answer is:
D. [tex]\(x = -25\)[/tex] or [tex]\(x = 7\)[/tex].
1. Positive Case:
- Assume [tex]\(x + 9\)[/tex] is positive or zero. The equation becomes:
[tex]\[
5(x + 9) = 80
\][/tex]
- Divide both sides by 5 to simplify:
[tex]\[
x + 9 = 16
\][/tex]
- Subtract 9 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 16 - 9 = 7
\][/tex]
2. Negative Case:
- Assume [tex]\(x + 9\)[/tex] is negative. Thus, we consider:
[tex]\[
5(-(x + 9)) = 80
\][/tex]
Which simplifies to:
[tex]\[
-5(x + 9) = 80
\][/tex]
- Divide both sides by -5:
[tex]\[
x + 9 = -16
\][/tex]
- Subtract 9 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = -16 - 9 = -25
\][/tex]
So, the solutions to the equation [tex]\(5|x+9|=80\)[/tex] are [tex]\(x = 7\)[/tex] or [tex]\(x = -25\)[/tex].
Therefore, the correct answer is:
D. [tex]\(x = -25\)[/tex] or [tex]\(x = 7\)[/tex].
