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Answer :
To solve the equation [tex]\( f(x) = 15 \)[/tex] for the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex], we need to find the values of [tex]\( x \)[/tex] that satisfy this equation. Let's go through the solution step-by-step:
1. Set the Function Equal to 15:
We start with the equation:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the Absolute Value:
Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide by 4:
Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x - 5| = 3
\][/tex]
4. Solve the Absolute Value Equation:
The equation [tex]\( |x - 5| = 3 \)[/tex] gives us two possible scenarios because the expression inside the absolute value can be either positive or negative:
Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = 8
\][/tex]
Case 2: [tex]\( x - 5 = -3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = 2
\][/tex]
5. Conclude the Solution:
So, the values of [tex]\( x \)[/tex] that satisfy the equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is [tex]\( x = 2, x = 8 \)[/tex].
1. Set the Function Equal to 15:
We start with the equation:
[tex]\[
4|x - 5| + 3 = 15
\][/tex]
2. Isolate the Absolute Value:
Subtract 3 from both sides to isolate the absolute value expression:
[tex]\[
4|x - 5| = 12
\][/tex]
3. Divide by 4:
Divide both sides by 4 to solve for the absolute value:
[tex]\[
|x - 5| = 3
\][/tex]
4. Solve the Absolute Value Equation:
The equation [tex]\( |x - 5| = 3 \)[/tex] gives us two possible scenarios because the expression inside the absolute value can be either positive or negative:
Case 1: [tex]\( x - 5 = 3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = 8
\][/tex]
Case 2: [tex]\( x - 5 = -3 \)[/tex]
- Add 5 to both sides:
[tex]\[
x = 2
\][/tex]
5. Conclude the Solution:
So, the values of [tex]\( x \)[/tex] that satisfy the equation are [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex].
Therefore, the correct answer is [tex]\( x = 2, x = 8 \)[/tex].
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