Answer :
Let’s solve the equation step by step.
We are given
[tex]$$
f(x)=4|x-5|+3,
$$[/tex]
and we need to find the values of [tex]$x$[/tex] for which [tex]$f(x)=15$[/tex]. That is,
[tex]$$
4|x-5|+3=15.
$$[/tex]
Step 1. Subtract [tex]$3$[/tex] from both sides to isolate the absolute value term:
[tex]$$
4|x-5| = 15-3 = 12.
$$[/tex]
Step 2. Divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]
Step 3. An absolute value equation [tex]$|x-5|=3$[/tex] has two cases:
- Case 1:
[tex]$$
x-5 = 3 \quad \Longrightarrow \quad x = 3+5 = 8.
$$[/tex]
- Case 2:
[tex]$$
x-5 = -3 \quad \Longrightarrow \quad x = -3+5 = 2.
$$[/tex]
Step 4. Therefore, the two solutions are:
[tex]$$
x=2 \quad \text{and} \quad x=8.
$$[/tex]
Comparing with the provided answer choices, the correct pair is [tex]$$\boxed{x=2,\ x=8}.$$[/tex]
We are given
[tex]$$
f(x)=4|x-5|+3,
$$[/tex]
and we need to find the values of [tex]$x$[/tex] for which [tex]$f(x)=15$[/tex]. That is,
[tex]$$
4|x-5|+3=15.
$$[/tex]
Step 1. Subtract [tex]$3$[/tex] from both sides to isolate the absolute value term:
[tex]$$
4|x-5| = 15-3 = 12.
$$[/tex]
Step 2. Divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5| = \frac{12}{4} = 3.
$$[/tex]
Step 3. An absolute value equation [tex]$|x-5|=3$[/tex] has two cases:
- Case 1:
[tex]$$
x-5 = 3 \quad \Longrightarrow \quad x = 3+5 = 8.
$$[/tex]
- Case 2:
[tex]$$
x-5 = -3 \quad \Longrightarrow \quad x = -3+5 = 2.
$$[/tex]
Step 4. Therefore, the two solutions are:
[tex]$$
x=2 \quad \text{and} \quad x=8.
$$[/tex]
Comparing with the provided answer choices, the correct pair is [tex]$$\boxed{x=2,\ x=8}.$$[/tex]
