3. Solve the equation $5 x+2|x|=3 x$

$(|x|=x$ when $x \geq 0$ and $|x|=-x$ when $x<0)$.

#
# Solution

Transform the equation to the form

$$
2 x+2|x|=0
$$

1) Solve the equation for $x \geq 0$.

For $x \geq 0$, we get

$$
\begin{gathered}
2 x+2 x=0 \\
4 x=0 \\
x=0
\end{gathered}
$$

The obtained value of $x$ satisfies the condition $x \geq 0$. Therefore, $x=0$ is not a solution to the equation.

2) Solve the equation for $x<0$.

For $x<0$, we get

$$
2 x-2 x=0
$$

The obtained equality is true for any value of $x$. Therefore, the solution will be any number from the considered interval, i.e., $x<0$.

Considering both cases, we get the solution to the equation: $x \leq 0$.

## Criteria

Only the first case is considered and correctly solved - 2 points.

Only the second case is considered and correctly solved - 3 points.

Both cases are considered, but the obtained equation is not solved in each of them - 1 point.

The first case is correctly solved, and in the second, the equation is obtained but no conclusion is made about its solution or the conclusion is incorrect - 3 points.

The second case is correctly solved, and in the first, the equation is obtained and solved, but the condition $x \geq 0$ is not checked, and the correct answer is obtained - 5 points.

A typo or arithmetic error that does not affect the solution process and the answer - 6 points.

If the solution is correct, but the answer is given in the form $x<0, x=0$, the task is fully credited - 7 points.