Problem 9. Solve the equation

$$
\log _{3} x=1+\frac{-3}{2 x-1}
$$

![](https://cdn.mathpix.com/cropped/2024_05_21_8debcc8c8285c4d79a3dg-079.jpg?height=548&width=485&top_left_y=286&top_left_x=340)
Solved. We construct the graphs of the functions $f(x)=\log _{3} x, \quad \varphi(x)=$ $=1+\frac{-3}{2 x-1}$ (Fig. 5).

On the half-interval $(-\infty ; 0]$, the function $f(x)$ is not defined, so there are no solutions to equation (1) on this interval.

On the interval $(0 ; 0.5)$, the function $\varphi(x)$ is positive, while the function $f(x)$ is negative, so there are no solutions to equation (1) here either.

At the point $x=0.5$, the function $\varphi(x)$ is not defined.

On the half-interval $(0.5 ; 1]$, the function $f(x)>-2$, and the function $\varphi(x)>0.4$.

Finally, on the interval $(3 ;+\infty)$, the function $\varphi(x)<1$.

Thus, equation (1) has no solutions.

## Task 8. IRRATIONAL EQUATIONS AND INEQUALITIES