Answer :
Sure, let's solve the equation [tex]\(5|x+9|=80\)[/tex] step by step.
1. Isolate the absolute value:
[tex]\[
|x+9| = \frac{80}{5} = 16
\][/tex]
2. Set up the cases for the absolute value equation:
When we have an equation with an absolute value, [tex]\( |x + 9| = 16 \)[/tex], it means that the expression inside can be either 16 or -16.
So we consider two separate cases:
- Case 1: [tex]\( x + 9 = 16 \)[/tex]
- Case 2: [tex]\( x + 9 = -16 \)[/tex]
3. Solve each case separately:
- Case 1:
[tex]\[
x + 9 = 16
\][/tex]
Subtract 9 from both sides:
[tex]\[
x = 16 - 9 = 7
\][/tex]
- Case 2:
[tex]\[
x + 9 = -16
\][/tex]
Subtract 9 from both sides:
[tex]\[
x = -16 - 9 = -25
\][/tex]
4. Write the set of solutions:
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(5|x+9|=80\)[/tex] are [tex]\(x = 7\)[/tex] and [tex]\(x = -25\)[/tex].
Given the answer choices:
- A. [tex]\(x=-7\)[/tex] or [tex]\(x=25\)[/tex]
- B. [tex]\(x=-25\)[/tex] or [tex]\(x=9\)[/tex]
- C. [tex]\(x=7\)[/tex] or [tex]\(x=16\)[/tex]
- D. [tex]\(x=-25\)[/tex] or [tex]\(x=7\)[/tex]
The correct answer is: D. [tex]\(x=-25\)[/tex] or [tex]\(x=7\)[/tex].
1. Isolate the absolute value:
[tex]\[
|x+9| = \frac{80}{5} = 16
\][/tex]
2. Set up the cases for the absolute value equation:
When we have an equation with an absolute value, [tex]\( |x + 9| = 16 \)[/tex], it means that the expression inside can be either 16 or -16.
So we consider two separate cases:
- Case 1: [tex]\( x + 9 = 16 \)[/tex]
- Case 2: [tex]\( x + 9 = -16 \)[/tex]
3. Solve each case separately:
- Case 1:
[tex]\[
x + 9 = 16
\][/tex]
Subtract 9 from both sides:
[tex]\[
x = 16 - 9 = 7
\][/tex]
- Case 2:
[tex]\[
x + 9 = -16
\][/tex]
Subtract 9 from both sides:
[tex]\[
x = -16 - 9 = -25
\][/tex]
4. Write the set of solutions:
The values of [tex]\(x\)[/tex] that satisfy the equation [tex]\(5|x+9|=80\)[/tex] are [tex]\(x = 7\)[/tex] and [tex]\(x = -25\)[/tex].
Given the answer choices:
- A. [tex]\(x=-7\)[/tex] or [tex]\(x=25\)[/tex]
- B. [tex]\(x=-25\)[/tex] or [tex]\(x=9\)[/tex]
- C. [tex]\(x=7\)[/tex] or [tex]\(x=16\)[/tex]
- D. [tex]\(x=-25\)[/tex] or [tex]\(x=7\)[/tex]
The correct answer is: D. [tex]\(x=-25\)[/tex] or [tex]\(x=7\)[/tex].
