Answer :
We start with the function
[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]
Step 1. Write the equation:
[tex]$$
4|x-5|+3=15.
$$[/tex]
Step 2. Subtract [tex]$3$[/tex] from both sides to isolate the absolute value term:
[tex]$$
4|x-5|=15-3,
$$[/tex]
which simplifies to
[tex]$$
4|x-5|=12.
$$[/tex]
Step 3. Divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5|=\frac{12}{4}=3.
$$[/tex]
Step 4. Solve the absolute value equation:
[tex]$$
|x-5|=3.
$$[/tex]
This gives us two cases:
Case 1:
[tex]$$
x-5=3 \quad \Longrightarrow \quad x=5+3=8.
$$[/tex]
Case 2:
[tex]$$
x-5=-3 \quad \Longrightarrow \quad x=5-3=2.
$$[/tex]
Thus, the two solutions are:
[tex]$$
x=2 \quad \text{and} \quad x=8.
$$[/tex]
Therefore, the values of [tex]$x$[/tex] for which [tex]$f(x)=15$[/tex] are [tex]$\boxed{2}$[/tex] and [tex]$\boxed{8}$[/tex].
[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]
Step 1. Write the equation:
[tex]$$
4|x-5|+3=15.
$$[/tex]
Step 2. Subtract [tex]$3$[/tex] from both sides to isolate the absolute value term:
[tex]$$
4|x-5|=15-3,
$$[/tex]
which simplifies to
[tex]$$
4|x-5|=12.
$$[/tex]
Step 3. Divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5|=\frac{12}{4}=3.
$$[/tex]
Step 4. Solve the absolute value equation:
[tex]$$
|x-5|=3.
$$[/tex]
This gives us two cases:
Case 1:
[tex]$$
x-5=3 \quad \Longrightarrow \quad x=5+3=8.
$$[/tex]
Case 2:
[tex]$$
x-5=-3 \quad \Longrightarrow \quad x=5-3=2.
$$[/tex]
Thus, the two solutions are:
[tex]$$
x=2 \quad \text{and} \quad x=8.
$$[/tex]
Therefore, the values of [tex]$x$[/tex] for which [tex]$f(x)=15$[/tex] are [tex]$\boxed{2}$[/tex] and [tex]$\boxed{8}$[/tex].