Find the largest rational number $p + q^2$ if $-1 < p \le 1.5$ and $-3 \le q \le 2$. Answer

1. Identify the maximum possible value of $p$.

Since $-1<p\leq 1.5$, the largest possible value of $p$ is
$p=1.5.$

2. Find the maximum possible value of $q^2$ and calculate the largest value of $p+q^2$.

Since $-3\leq q\leq 2$, the value of $q$ with the greatest magnitude is $q=-3$.

Hence
$q^2=(-3)^2=9.$

So the largest value is
$p+q^2=1.5+9=10.5.$