Example 9. Solve the inequality

$$
\frac{\sqrt{2-x}+4 x-3}{x} \geqslant 2
$$

## 148 CH. 3. INEQUALITIES WITH ONE UNKNOWN
Solution. From the conditions $2-x \geqslant 0$ and $x \neq 0$, we find the domain of the inequality (11): $x<2$, and $x \neq 0$. If $x>1$, then by the properties of exponents we have

$$
\sqrt{2-x} 0 , } \\
{ \sqrt { 2 - x } \geqslant 3 - 2 x , } \\
{ \{ \begin{array} { l } 
{ x 0, \\
3-2 x0, \\
3-2 x \geqslant 0, \\
2-x \geqslant(3-2 x)^{2},
\end{array}\right. \\
\begin{array}{l}
x 3 / 2 , } \\
{ \{ } \\
{ 0 \sqrt{2}, \\
t \leqslant 1
\end{array}, \Leftrightarrow\left[\begin{array} { l } 
{ t \geqslant 0 , } \\
{ t > \sqrt { 2 } } \\
{ \{ \begin{array} { l } 
{ t \geqslant 0 } \\
{ t \leqslant 1 }
\end{array} }
\end{array} \Leftrightarrow \left[\begin{array}{l}
0 \leqslant t \leqslant 1 \\
t>\sqrt{2}
\end{array}\right.\right.\right.
\end{aligned}
$$

## 150 CH. 3. INEQUALITIES WITH ONE UNKNOWN

Thus, equation (11) is equivalent to the system of inequalities

$$
\begin{aligned}
& {\left[\begin{array} { l } 
{ 0 \leqslant \sqrt { 2 - x } \leqslant 1 , } \\
{ \sqrt { 2 - x } > \sqrt { 2 } }
\end{array} \Leftrightarrow \left[\begin{array}{c}
0 \leqslant 2-x \leqslant 1, \\
2-x>2
\end{array} \Leftrightarrow\right.\right.} \\
& \Leftrightarrow\left[\begin{array} { c } 
{ - 2 \leqslant - x \leqslant - 1 , } \\
{ - x > 0 }
\end{array} \Leftrightarrow \left[\begin{array}{c}
1 \leqslant x \leqslant 2, \\
x<0 .
\end{array}\right.\right.
\end{aligned}
$$

Thus, the set of all solutions to the inequality (11) is the set $(-\infty ; 0) \cup[1 ; 2]$.