The false alarm probability $\alpha$ (also called the Type I
error probability) is the probability that the value of the single
observation $X$ lies in the rejection region $R$ when $H_0$ is the true
hypothesis. Note that when $X \in R$, the null hypothesis
$H_0$ is rejected
(and in this case, the alternative $H_1$ is accepted).
So, (for continuous random variables)
$$0.1 \geq \alpha = \int_R f(x\mid \theta_0)\,\mathrm dx.\tag{1}$$
Any region for which $(1)$ holds is a possible candidate that can
serve as the rejection region. Which of these should we choose?
Well, the power $\beta$ of the test is the probability of correctly
declaring in favor of $H_1$ (rejecting $H_0$) when $H_1$ is indeed
the true hypothesis, that is,
$$\beta = \int_R f(x\mid \theta_1)\,\mathrm dx
= \int_R f(x\mid \theta_0)\times
\frac{f(x\mid \theta_1)}{f(x\mid \theta_0)}\,\mathrm dx
= \int_R f(x\mid \theta_0)\Lambda(x)\,\mathrm dx\tag{2}$$
where $\Lambda(x)$ is the likelihood ratio. Maximizing
the power $\beta$ while still managing to avoid the abyss
of $\alpha$ exceeding $0.1$ seems like a reasonable goal to
strive for. But how should we go about doing this?
Clearly, we want to include in $R^*$, the optimum region
for which $(1)$ holds and $\beta$ is maximized, those $x$
for which $\Lambda(x)$ is as large as possible. So,
start with the global maximum of $\Lambda(x)$ as the
point we definitely want to include in $R^*$ and then
search for points where $\Lambda(x)$ is smaller than the
global maximum but still large, including such points in $R^*$.
Keep your eye on $(1)$ though, and stop as soon as you
have achieved the desired $\alpha$. The region $R^*$
depends on what the likelihood ratio looks like. It is
not the case that $R^* = \{x\colon x \geq k\}$ as you think
but rather that
$$R^* = \left\{x\colon \Lambda(x) \geq k\right\}.$$
This translates into $\{x\colon x\geq \hat{k}\}$ only if
$\Lambda(x)$ is an increasing function of $x$.
