Answer :
To solve the equation [tex]\( f(x) = 15 \)[/tex] for the function given by [tex]\( f(x) = 4|x-5| + 3 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that make the equation true:
1. Start with the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x-5| = 12
\][/tex]
3. Divide both sides by 4 to further simplify:
[tex]\[
|x-5| = 3
\][/tex]
4. At this point, we need to solve for [tex]\( x \)[/tex] in two separate cases because of the absolute value:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = -3 + 5 = 2
\][/tex]
So, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex]. Therefore, the correct answer is:
[tex]\( x = 2, x = 8 \)[/tex]
1. Start with the equation:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x-5| = 12
\][/tex]
3. Divide both sides by 4 to further simplify:
[tex]\[
|x-5| = 3
\][/tex]
4. At this point, we need to solve for [tex]\( x \)[/tex] in two separate cases because of the absolute value:
- Case 1: [tex]\( x-5 = 3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- Case 2: [tex]\( x-5 = -3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = -3 + 5 = 2
\][/tex]
So, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex]. Therefore, the correct answer is:
[tex]\( x = 2, x = 8 \)[/tex]
