Why $\operatorname{gap-A}[1.5a, 1.5b + 1/4]$ has an efficient solution?
Given an instance $x$ of $\mathsf{gap\text{-}A}(1.5a,1.5b+1/4)$, calculate $y_x$ using the $a/b$ approximation algorithm, so that
$$
\frac{a}{b} opt(x) \leq val(y_x) \leq opt(x).
$$
If $opt(x) \geq 1.5b+1/4$ then
$$
val(y_x) \geq \frac{a}{b} (1.5b+1/4) &gt; 1.5a.
$$
Conversely, if $opt(x) \leq 1.5a$ then
$$
val(y_x) \leq 1.5a.
$$
You take it from here.