8. Find the sum of all integer values of $\mathrm{h}$ for which the equation ||$r+h|-r|-4 r=9|r-3|$ in terms of $r$ has no more than one solution.

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# Solution

Consider the function $\mathrm{f}(\mathrm{r})=9|\mathrm{r}-3|-|| \mathrm{r}+\mathrm{h}|-\mathrm{r}|+4 \mathrm{r}$. The coefficient of the first modulus in absolute value is greater than the sum of the other coefficients of $\mathrm{r}$. $9>1+1+4$. Therefore, on all intervals up to $r=3$, the coefficient of the linear increment is negative, and after $\mathrm{r}=3$ it is positive. $\mathrm{r}=3$ is a point of minimum. For the equation $\mathrm{f}(\mathrm{r})=0$ to have no more than one root, it is necessary and sufficient that the inequality:

$$
\mathrm{f}(3) \geq 0
$$

$\Rightarrow$ Let $|\mathrm{h}+3|=\mathrm{t} ; 12-|t-3| \geq 0$

$(\mathrm{t}-3)^{2}-12^{2} \leq 0$

$(\mathrm{t}-15)(\mathrm{t}+9) \leq 0$

$\mathrm{t} \in[-9 ; 15]$

$|\mathrm{h}+3| \leq 15$

Answer: -93.