Answer :
We are given 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. Set up the equation:
[tex]$$
4|x-5|+3=15.
$$[/tex]
Step 2. Subtract [tex]$3$[/tex] from both sides to isolate the term with the absolute value:
[tex]$$
4|x-5|=15-3=12.
$$[/tex]
Step 3. Divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5|=\frac{12}{4}=3.
$$[/tex]
Step 4. The equation [tex]$|x-5|=3$[/tex] means that the expression inside the absolute value, [tex]$x-5$[/tex], can be either [tex]$3$[/tex] or [tex]$-3$[/tex]. So, we have two cases:
1) [tex]$$x-5=3 \quad \Rightarrow \quad x=3+5=8,$$[/tex]
2) [tex]$$x-5=-3 \quad \Rightarrow \quad x=-3+5=2.$$[/tex]
Thus, the two solutions are [tex]$x=2$[/tex] and [tex]$x=8$[/tex].
Final Answer: The values of [tex]$x$[/tex] that satisfy [tex]$f(x)=15$[/tex] are [tex]$$x=2 \quad \text{and} \quad x=8.$$[/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. Set up the equation:
[tex]$$
4|x-5|+3=15.
$$[/tex]
Step 2. Subtract [tex]$3$[/tex] from both sides to isolate the term with the absolute value:
[tex]$$
4|x-5|=15-3=12.
$$[/tex]
Step 3. Divide both sides by [tex]$4$[/tex]:
[tex]$$
|x-5|=\frac{12}{4}=3.
$$[/tex]
Step 4. The equation [tex]$|x-5|=3$[/tex] means that the expression inside the absolute value, [tex]$x-5$[/tex], can be either [tex]$3$[/tex] or [tex]$-3$[/tex]. So, we have two cases:
1) [tex]$$x-5=3 \quad \Rightarrow \quad x=3+5=8,$$[/tex]
2) [tex]$$x-5=-3 \quad \Rightarrow \quad x=-3+5=2.$$[/tex]
Thus, the two solutions are [tex]$x=2$[/tex] and [tex]$x=8$[/tex].
Final Answer: The values of [tex]$x$[/tex] that satisfy [tex]$f(x)=15$[/tex] are [tex]$$x=2 \quad \text{and} \quad x=8.$$[/tex]
