The line
$x - 2y - 1 = 0 \tag{1}$
may also be written
$y = \dfrac{1}{2}x - \dfrac{1}{2}; \tag{2}$
from this we see its slope is $\frac{1}{2}$. Thus the slope of a line normal or perpendicular to this line is $-2$.
For any value of $x$ other than $\frac{1}{2}$, the slope of the curve
$y(x)= \dfrac{1}{2x - 1} = (2x - 1)^{-1} \tag {3}$
is given by
$y'(x) = -2(2x - 1)^{-2}, \tag{4}$
We seek the points on the curve (3) where the slope is $-2$, yielding
$-2(2x - 1)^{-2} = -2, \tag{5}$
or
$(2x - 1)^{-2} = 1, \tag{6}$
or
$(2x - 1)^2 = 1; \tag{7}$
thus
$2x - 1 = \pm 1, \tag{8}$
whence
$x = 0, 1. \tag{9}$
That
$y'(0) = -2, y'(1) = -2 \tag{10}$
is easily checked using (4); thus the slope of the curve (3) is normal to the line (2) at the points $(0, -1)$ and $(1, 1)$.
Note added in edit: it will be observed that the value of $b$, that is, the $y$-intercept of the tangent lines to (3), does not enter in to the above calculations, by reason of the fact that the problem does not call for these lines, only for certain points on the given curve. However, having found these points where the slope is $-2$, the corresponding $b$-values are easily discovered. Each of these lines is of slope $-2$, and is represented by an equation of the form
$y = -2x + b; \tag{11}$
if $(0, -1)$ lies on such a line, we must have
$-1 = -2 \cdot 0 + b, \tag{12}$
or
$b = -1; \tag{13}$
the other $b$-value satisfies
$1 = -2 \cdot 1 + b, \tag{14}$
whence
$b = 3. \tag{15}$