Answer :
We are given the function
[tex]$$
f(x) = 4|x-5| + 3,
$$[/tex]
and we wish to find all values of [tex]$x$[/tex] such that
[tex]$$
f(x) = 15.
$$[/tex]
Step 1. Set up the equation:
[tex]$$
4|x-5| + 3 = 15.
$$[/tex]
Step 2. Isolate the absolute value term:
Subtract [tex]$3$[/tex] from both sides:
[tex]$$
4|x-5| = 15 - 3 = 12.
$$[/tex]
Then, divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]
Step 3. Solve the absolute value equation:
An equation of the form
[tex]$$
|x-5| = 3
$$[/tex]
has two cases:
1. Case 1: [tex]$x - 5 = 3$[/tex]
Add [tex]$5$[/tex] to both sides:
[tex]$$
x = 3 + 5 = 8.
$$[/tex]
2. Case 2: [tex]$x - 5 = -3$[/tex]
Add [tex]$5$[/tex] to both sides:
[tex]$$
x = -3 + 5 = 2.
$$[/tex]
Conclusion:
The values of [tex]$x$[/tex] for which [tex]$f(x) = 15$[/tex] are
[tex]$$
x = 2 \quad \text{and} \quad x = 8.
$$[/tex]
Thus, the correct answer is: [tex]$x=2, x=8$[/tex].
[tex]$$
f(x) = 4|x-5| + 3,
$$[/tex]
and we wish to find all values of [tex]$x$[/tex] such that
[tex]$$
f(x) = 15.
$$[/tex]
Step 1. Set up the equation:
[tex]$$
4|x-5| + 3 = 15.
$$[/tex]
Step 2. Isolate the absolute value term:
Subtract [tex]$3$[/tex] from both sides:
[tex]$$
4|x-5| = 15 - 3 = 12.
$$[/tex]
Then, divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]
Step 3. Solve the absolute value equation:
An equation of the form
[tex]$$
|x-5| = 3
$$[/tex]
has two cases:
1. Case 1: [tex]$x - 5 = 3$[/tex]
Add [tex]$5$[/tex] to both sides:
[tex]$$
x = 3 + 5 = 8.
$$[/tex]
2. Case 2: [tex]$x - 5 = -3$[/tex]
Add [tex]$5$[/tex] to both sides:
[tex]$$
x = -3 + 5 = 2.
$$[/tex]
Conclusion:
The values of [tex]$x$[/tex] for which [tex]$f(x) = 15$[/tex] are
[tex]$$
x = 2 \quad \text{and} \quad x = 8.
$$[/tex]
Thus, the correct answer is: [tex]$x=2, x=8$[/tex].
