8. Find the sum of all integer values of c for which the equation $15|p-1|+|3 p-| p+c||=4$ has at least one root with respect to p

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# Solution:

Consider the function $\mathfrak{1} \mathbb{R}=15|\mathrm{p}-1|+|3 \mathrm{p}-| \mathrm{p}+\mathrm{c}||-4 \mathrm{p}$. The coefficient of the first modulus in absolute value is greater than the sum of the other coefficients of $\mathrm{p}$. $15>4+1+3$. Therefore, on all intervals up to $\mathrm{p}=1$, the coefficient of the linear increment is negative, and after $\mathrm{p}=1$ it is positive. $\mathrm{P}=1$ is a point of minimum. For the equation $\mathrm{f}(\mathrm{p})=0$ to have at least one root, it is necessary and sufficient that the inequality:

$\mathrm{f}(1) \leq 0$

$\Rightarrow$ Let $|\mathrm{c}+1|=\mathrm{t} ;|3-\mathrm{t}|-4 \leq 0$

$(3-t)^{2}-4^{2} \geq 0$

$(7-t)(-1-t) \geq 0$

$\mathrm{t} \in[-1 ; 7]$

$|c+1| \leq 7$

Answer: -15.