1. Find all solutions to the equation

$$
1=\frac{|3 x-7|-|9-2 x|}{|x+2|}
$$
1. From the assignment, it follows that $x \neq -2$. After multiplying by the expression $|x+2|$, we get the equation $|x+2|=|3x-7|-|9-2x|$, which we will now solve.

The zeros of the three absolute values with the unknown divide the real axis into four intervals, in which each of the corresponding binomials has the same sign, while at the boundary points it is zero. In each of these intervals, we can therefore solve the corresponding equation without absolute values.

$\triangleright x \in (-\infty, -2)$: we get the equation $-x-2=-3x+7-(9-2x)$, which after simplification becomes the identity $0=0$. All numbers from the examined interval satisfy the original equation.

$\triangleright x \in \left\langle -2, \frac{7}{3} \right)$: we get the equation $x+2=-3x+7-(9-2x)$ or $2x=-4$ with the only solution $x=-2$, which, as we already know, does not satisfy the original equation.

$\triangleright x \in \left\langle \frac{7}{3}, \frac{9}{2} \right)$: we get the equation $x+2=3x-7-(9-2x)$ with the solution $x=\frac{9}{2}$, which does not lie in the considered interval.

$\triangleright x \in \left\langle \frac{9}{2}, \infty \right)$: we get the equation $x+2=3x-7-(-9+2x)$, which after simplification becomes the identity $0=0$. All $x$ from this interval satisfy the equation.

Conclusion. All solutions to the problem form the set $(-\infty, -2) \cup \left\langle \frac{9}{2}, \infty \right)$.

For a complete solution, award 6 points. For the correct approach (effort to remove absolute values), award 1 point. For investigating the equation in each of the intervals, award 1 point. Conclusion (explicitly stating the set of solutions) 1 point. For each incorrectly calculated zero point, deduct 2 points (however, no more than 5 points). The equation can also be worked with in its original form with the unremoved fraction.