Given the inequality \(3y + 5x \leq 15\) and the conditions \(y > 0\), \(y < 3\), and \(x > 0\), let's find the integral values of \(x\) and \(y\) that satisfy these conditions.

First, let's consider the range for \(y\). The conditions \(y > 0\) and \(y < 3\) mean that \(y\) must be a positive integer less than 3. Therefore, \(y\) can take the values 1 and 2.

Next, let's consider the range for \(x\). The condition \(x > 0\) means that \(x\) must be a positive integer.

Now, let's substitute the values of \(x\) and \(y\) into the inequality to see which pairs satisfy \(3y + 5x \leq 15\):

For \(y = 1\):

\(3(1) + 5x \leq 15\)

\(3 + 5x \leq 15\)

\(5x \leq 12\)

\(x \leq 2.4\)

For \(y = 2\):

\(3(2) + 5x \leq 15\)

\(6 + 5x \leq 15\)

\(5x \leq 9\)

\(x \leq 1.8\)

Since \(x\) must be a positive integer, the possible values for \(x\) are 1 and 2.

Therefore, the integral values of \(x\) and \(y\) that satisfy the conditions are:

(1, 1), (1, 2), (2, 1).