Question

$\sqrt{|1 - x|} = kx$

For tangency of two curves

$$\begin{matrix} & \sqrt{x_{1} - 1} = {kx}_{1} \\ \text{~and~} & \frac{1}{2\sqrt{x_{1} - 1}} = k \\ \Rightarrow & \sqrt{x_{1} - 1} = \frac{x_{1}}{2\sqrt{x_{1} - 1}} \\ \Rightarrow & 2\left( x_{1} - 1 \right) = x_{1} \Rightarrow x_{1} = 2,k = \frac{1}{2} \end{matrix}$$

From graph it is clear that equation has

$$\begin{matrix} \text{~one solution if~}k & \ \in ( - \infty,0\rbrack \cup \left( \frac{1}{2},\infty \right) \\ \text{~two solutions if~}k & \ = \frac{1}{2} \\ \text{~three solutions if~}k & \ \in \left( 0,\frac{1}{2} \right) \end{matrix}$$