Answer :
To solve the equation [tex]\(5|x+9| = 80\)[/tex], we need to isolate the absolute value and then consider the two possible scenarios for the expression within the absolute value.
1. Isolate the Absolute Value:
Divide both sides of the equation by 5:
[tex]\[
|x+9| = \frac{80}{5} = 16
\][/tex]
2. Consider the Two Cases:
An absolute value equation [tex]\( |A| = B \)[/tex] implies two cases:
- Case 1: [tex]\( A = B \)[/tex]
- Case 2: [tex]\( A = -B \)[/tex]
Apply these cases to our specific equation, [tex]\( |x+9| = 16 \)[/tex]:
- Case 1:
[tex]\[
x + 9 = 16
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 16 - 9 = 7
\][/tex]
- Case 2:
[tex]\[
x + 9 = -16
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -16 - 9 = -25
\][/tex]
3. Write the Final Solution:
Therefore, the solutions to the equation [tex]\(5|x+9| = 80\)[/tex] are [tex]\(x = -25\)[/tex] or [tex]\(x = 7\)[/tex].
So the correct answer is D. [tex]\( x = -25 \)[/tex] or [tex]\( x = 7 \)[/tex].
1. Isolate the Absolute Value:
Divide both sides of the equation by 5:
[tex]\[
|x+9| = \frac{80}{5} = 16
\][/tex]
2. Consider the Two Cases:
An absolute value equation [tex]\( |A| = B \)[/tex] implies two cases:
- Case 1: [tex]\( A = B \)[/tex]
- Case 2: [tex]\( A = -B \)[/tex]
Apply these cases to our specific equation, [tex]\( |x+9| = 16 \)[/tex]:
- Case 1:
[tex]\[
x + 9 = 16
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = 16 - 9 = 7
\][/tex]
- Case 2:
[tex]\[
x + 9 = -16
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
x = -16 - 9 = -25
\][/tex]
3. Write the Final Solution:
Therefore, the solutions to the equation [tex]\(5|x+9| = 80\)[/tex] are [tex]\(x = -25\)[/tex] or [tex]\(x = 7\)[/tex].
So the correct answer is D. [tex]\( x = -25 \)[/tex] or [tex]\( x = 7 \)[/tex].
