1. Solve the equation

$$
\sqrt{5 x+1}-2 \sqrt{4-x}+\sqrt{5} \sqrt{x+2}=\sqrt{61-4 x}
$$
Solution. The given equation is equivalent to the equation

$$
\sqrt{5 x+1}+\sqrt{5 x+10}=\sqrt{61-4 x}+\sqrt{16-4 x}
$$

where all expressions are defined for $-\frac{1}{5} \leq x \leq 4$. Let us denote

$$
f(x)=\sqrt{5 x+1}+\sqrt{5 x+10}, g(x)=\sqrt{61-4 x}+\sqrt{16-4 x}
$$

Then $f(3)=g(3)=9$. For $-\frac{1}{5} \leq x < 3$, we have $f(x) < f(3) = g(3) > g(x)$. For $3 < x \leq 4$, we have $f(x) > f(3) = g(3) < g(x)$. Therefore, $x=3$ is the only solution to the equation.