Answer :
To solve the problem of finding the values of [tex]\( x \)[/tex] for which the function [tex]\( f(x) = 4|x-5| + 3 \)[/tex] equals 15, we can follow these steps:
1. Set up the equation: Start with the equation of the function set equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression: Subtract 3 from both sides to get:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value: Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Consider the two cases of the absolute value equation: The equation [tex]\(|x-5| = 3\)[/tex] results in two possible equations:
- Case 1: [tex]\(x - 5 = 3\)[/tex]
- Case 2: [tex]\(x - 5 = -3\)[/tex]
5. Solve each case:
- For the first case [tex]\(x - 5 = 3\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For the second case [tex]\(x - 5 = -3\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].
1. Set up the equation: Start with the equation of the function set equal to 15:
[tex]\[
4|x-5| + 3 = 15
\][/tex]
2. Isolate the absolute value expression: Subtract 3 from both sides to get:
[tex]\[
4|x-5| = 12
\][/tex]
3. Solve for the absolute value: Divide both sides by 4:
[tex]\[
|x-5| = 3
\][/tex]
4. Consider the two cases of the absolute value equation: The equation [tex]\(|x-5| = 3\)[/tex] results in two possible equations:
- Case 1: [tex]\(x - 5 = 3\)[/tex]
- Case 2: [tex]\(x - 5 = -3\)[/tex]
5. Solve each case:
- For the first case [tex]\(x - 5 = 3\)[/tex]:
[tex]\[
x = 3 + 5 = 8
\][/tex]
- For the second case [tex]\(x - 5 = -3\)[/tex]:
[tex]\[
x = -3 + 5 = 2
\][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = 15 \)[/tex] are [tex]\( x = 8 \)[/tex] and [tex]\( x = 2 \)[/tex].
