Answer :
To find the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x - 5| + 3 \)[/tex] equals 15, you can follow these steps:
1. Set up the equation:
Begin with the expression for the function:
[tex]\[
f(x) = 4|x - 5| + 3
\][/tex]
We want to find [tex]\( x \)[/tex] values such that:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value:
Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. Consider both cases for the absolute value:
The equation [tex]\( |x - 5| = 3 \)[/tex] implies two possible cases:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = 3 \quad \Rightarrow \quad x = 3 + 5 = 8
\][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = -3 \quad \Rightarrow \quad x = -3 + 5 = 2
\][/tex]
5. Identify the solution:
The values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is [tex]\( x = 2, x = 8 \)[/tex].
1. Set up the equation:
Begin with the expression for the function:
[tex]\[
f(x) = 4|x - 5| + 3
\][/tex]
We want to find [tex]\( x \)[/tex] values such that:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the absolute value:
Subtract 3 from both sides to isolate the absolute value term:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Solve for the absolute value:
Divide both sides by 4:
[tex]\[
|x - 5| = 3
\][/tex]
4. Consider both cases for the absolute value:
The equation [tex]\( |x - 5| = 3 \)[/tex] implies two possible cases:
- Case 1: [tex]\( x - 5 = 3 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = 3 \quad \Rightarrow \quad x = 3 + 5 = 8
\][/tex]
- Case 2: [tex]\( x - 5 = -3 \)[/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x - 5 = -3 \quad \Rightarrow \quad x = -3 + 5 = 2
\][/tex]
5. Identify the solution:
The values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is [tex]\( x = 2, x = 8 \)[/tex].
