Example 3. Find all roots of the equation

$$
2\left|x^{2}+2 x-5\right|=x-1
$$

satisfying the inequality $x<\sqrt{2}$.
Solution. The given equation is equivalent to the following system of equations

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-037.jpg?height=134&width=1133&top_left_y=624&top_left_x=470)

which on the set $(-\infty ; \sqrt{2})$ is equivalent to the following system

$$
\left\{\begin{array} { l } 
{ 2 ( x ^ { 2 } + 2 x - 5 ) = x - 1 , } \\
{ 1 \leqslant x < \sqrt{2} }
\end{array}\right.
$$

For $x > 1$, we have the following chain of numerical inequalities: $\sqrt{113} / 4 > 9 / 4 \Leftrightarrow \sqrt{113} > 9 \Leftrightarrow 113 > 81$.

The last numerical inequality is true, and therefore the original inequality is also true.

From the inequality $\sqrt{113} / 4 - 5 / 4 \geqslant \sqrt{2}$, we have $\sqrt{113} > 4 \sqrt{2} + 5 \Leftrightarrow 113 \geqslant 32 + 40 \sqrt{2} + 25 \Leftrightarrow 56 \geqslant 40 \sqrt{2} \Leftrightarrow 7 \geqslant 5 \sqrt{2} \Leftrightarrow 49 \geqslant 50$. The last inequality is false, and therefore the original inequality is also false; hence, $\frac{\sqrt{113} - 5}{4}$ is less than $\sqrt{2}$. Thus, the condition $1 \leqslant \frac{\sqrt{113} - 5}{4} < \sqrt{2}$ is satisfied, and therefore the number $\frac{\sqrt{113} - 5}{4}$ is a solution to the second system.

The number $\frac{-5 - \sqrt{113}}{4}$ is negative, and therefore it does not belong to the interval $[1 ; \sqrt{2})$ and is not a solution to the system. Thus, the second system of the combined system has a unique root $\frac{\sqrt{113} - 5}{4}$.

Therefore, the combined system of the two systems, and consequently the original equation, has a unique root, the number $\frac{\sqrt{113} - 5}{4}$.