9. For what values of the parameter $\boldsymbol{a}$ does the equation $f(x)=p(x)$ have one solution, if $f(x)=\left|\frac{2 x^{3}-x^{2}-18 x+9}{(1.5 x+1)^{2}-(0.5 x-2)^{2}}\right|, p(x)=|-2 x+2|+a$. If there are more than one value of the parameter, indicate their sum in the answer.
# Solution.

Simplify $f(x)=\left|\frac{2 x^{3}-x^{2}-18 x+9}{(1.5 x+1)^{2}-(0.5 x-2)^{2}}\right|$, we get $f(x)=|x-3|$, where $x \neq 0.5, x \neq -3$.

Solve the equation $|x-3|=|-2 x+2|+a$, where $x \neq 0.5, x \neq -3$ graphically in the system $x O a$.

1) $\left\{\begin{array}{c}x<1 \\ -x+3=-2 x+2+a\end{array} \quad\left\{\begin{array}{c}x<1 \\ a=x+1\end{array}\right.\right.$
2) $\left\{\begin{array}{c}1 \leq x<3 \\ -x+3=2 x-2+a\end{array} \quad\left\{\begin{array}{c}1 \leq x<3 \\ a=-3 x+5\end{array}\right.\right.$
3) $\left\{\begin{array}{c}x \geq 3 \\ x-3=2 x-2+a\end{array} \quad\left\{\begin{array}{c}x \geq 3 \\ a=-x-1\end{array}\right.\right.$

The equation has one solution when $a=-2, a=2, a=1.5$.

![](https://cdn.mathpix.com/cropped/2024_05_06_de39ee82fbf77ee5c736g-7.jpg?height=774&width=708&top_left_y=824&top_left_x=1234)

The sum is 1.5

Answer: 1.5.