Answer :
Let's solve the equation [tex]\( f(x) = 15 \)[/tex] for the function [tex]\( f(x) = 4|x-5|+3 \)[/tex].
First, set the function equal to 15:
[tex]\[ 4|x-5| + 3 = 15 \][/tex]
Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[ 4|x-5| = 12 \][/tex]
Divide both sides by 4:
[tex]\[ |x-5| = 3 \][/tex]
The equation [tex]\( |x-5| = 3 \)[/tex] means that the expression inside the absolute value, [tex]\( x-5 \)[/tex], can be either 3 or -3. Let's solve for [tex]\( x \)[/tex] in both cases:
1. [tex]\( x-5 = 3 \)[/tex]
[tex]\[
x = 3 + 5 = 8
\][/tex]
2. [tex]\( x-5 = -3 \)[/tex]
[tex]\[
x = -3 + 5 = 2
\][/tex]
So the solutions for [tex]\( x \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is [tex]\( x=2, x=8 \)[/tex].
First, set the function equal to 15:
[tex]\[ 4|x-5| + 3 = 15 \][/tex]
Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[ 4|x-5| = 12 \][/tex]
Divide both sides by 4:
[tex]\[ |x-5| = 3 \][/tex]
The equation [tex]\( |x-5| = 3 \)[/tex] means that the expression inside the absolute value, [tex]\( x-5 \)[/tex], can be either 3 or -3. Let's solve for [tex]\( x \)[/tex] in both cases:
1. [tex]\( x-5 = 3 \)[/tex]
[tex]\[
x = 3 + 5 = 8
\][/tex]
2. [tex]\( x-5 = -3 \)[/tex]
[tex]\[
x = -3 + 5 = 2
\][/tex]
So the solutions for [tex]\( x \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is [tex]\( x=2, x=8 \)[/tex].
