Answer :
Certainly! Let's solve the equation [tex]\( f(x) = 15 \)[/tex] for the given function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex].
We start with the equation:
[tex]\[ f(x) = 4|x - 5| + 3 \][/tex]
We want to find where [tex]\( f(x) = 15 \)[/tex], so we set the equation equal to 15:
[tex]\[ 4|x - 5| + 3 = 15 \][/tex]
Next, we subtract 3 from both sides to isolate the absolute value term:
[tex]\[ 4|x - 5| = 12 \][/tex]
Then, we divide both sides by 4:
[tex]\[ |x - 5| = 3 \][/tex]
The absolute value equation [tex]\( |x - 5| = 3 \)[/tex] gives us two possible equations to solve:
1. [tex]\( x - 5 = 3 \)[/tex]
2. [tex]\( x - 5 = -3 \)[/tex]
Now, let's solve these equations one by one:
1. For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[ x = 3 + 5 \][/tex]
[tex]\[ x = 8 \][/tex]
2. For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
So, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 15 \)[/tex] are:
[tex]\[ x = 2 \quad \text{and} \quad x = 8 \][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x=2, x=8}
\][/tex]
We start with the equation:
[tex]\[ f(x) = 4|x - 5| + 3 \][/tex]
We want to find where [tex]\( f(x) = 15 \)[/tex], so we set the equation equal to 15:
[tex]\[ 4|x - 5| + 3 = 15 \][/tex]
Next, we subtract 3 from both sides to isolate the absolute value term:
[tex]\[ 4|x - 5| = 12 \][/tex]
Then, we divide both sides by 4:
[tex]\[ |x - 5| = 3 \][/tex]
The absolute value equation [tex]\( |x - 5| = 3 \)[/tex] gives us two possible equations to solve:
1. [tex]\( x - 5 = 3 \)[/tex]
2. [tex]\( x - 5 = -3 \)[/tex]
Now, let's solve these equations one by one:
1. For [tex]\( x - 5 = 3 \)[/tex]:
[tex]\[ x = 3 + 5 \][/tex]
[tex]\[ x = 8 \][/tex]
2. For [tex]\( x - 5 = -3 \)[/tex]:
[tex]\[ x = -3 + 5 \][/tex]
[tex]\[ x = 2 \][/tex]
So, the values of [tex]\( x \)[/tex] that satisfy [tex]\( f(x) = 15 \)[/tex] are:
[tex]\[ x = 2 \quad \text{and} \quad x = 8 \][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x=2, x=8}
\][/tex]