4. Find all solutions to the equation $x=|2 x-| 60-2 x||$.

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$\mathbf{I} / 4$. Let $f(x)=|2 x-| 60-2 x||$. Since the expression $|2 x-| 60-2 x||$ only takes non-negative values, a negative $x$ cannot solve the equation $x=f(x)$.

First, let's consider the case $60-2 x \geq 0$, which means $x \leq 30$. Then the equation can be written as: $x=|2 x-60+2 x|$ or $x=|4 x-60|$. We will consider 2 cases. If $4 x-60 \geq 0$, then $x \geq 15$, and the equation simplifies to $x=4 x-60$, which has the solution $x=20$. If $4 x-60 < 0$, then $x < 15$, and the equation simplifies to $x=60-4 x$, which has the solution $x=12$.

Next, let's consider the case $60-2 x < 0$, which means $x > 30$. Then the equation can be written as: $x=|2 x+60-2 x|$ or $x=60$, which is

![](https://cdn.mathpix.com/cropped/2024_06_07_af92d6d33b11d7748e87g-07.jpg?height=563&width=563&top_left_y=712&top_left_x=1272)
another solution.

Analysis of cases based on inner and outer absolute value: $2+2=4$ points. All solutions: $1+1+1=3$ points.