7. For what values of the parameter $a$ will the minimum value of the function

$$
f(x)=|7 x-3 a+8|+|5 x+4 a-6|+|x-a-8|-24
$$

be the smallest.
Solution. If $7 x>3 a-8$, then with any method of unfolding the remaining two modules, the coefficient of $x$ will be positive, meaning the function will be increasing. If, however, $7 x<3 a-8$, then with any method of unfolding the modules, the coefficient of $x$ will be negative and the function will be decreasing. Thus, $x=\frac{3 a-8}{7}$ is a point of global minimum and

$$
f_{\min }(a)=\left|5 \cdot \frac{3 a-8}{7}+4 a-6\right|+\left|\frac{3 a-8}{7}-a-8\right|-24=\frac{|43 a-82|+|4 a-65|}{7}-24
$$

We conduct reasoning similar to that provided above and find that the minimum value of this function is achieved at $a=\frac{82}{43}$.

Answer: $a=\frac{82}{43}$.