Assumption of sufficient wealth in quasi-linear preferences
This depends on the exact preferences, but usually the utility function
$$
U(x,y) = v(x) + y
$$
is such that
$$
\lim_{x \to 0} |MRS(x,y)| = \lim_{x \to 0} \frac{\text{d}v(x)}{\text{d} x} = \infty.
$$
In this case it is the consumption of an additional marginal unit of the nonlinear good $x$ that is infinitely useful compared to the consumption of an additional marginal unit of the linear good $y$. Hence the consumer will consume some units of $x$ before buying any of $y$. If the consumer is not 'wealthy enough', the corner solution will be such that $x &gt; 0$, $y = 0$.