Mathematics

The key insight is that two points lie on opposite sides of a line if and only if they produce opposite signs when you substitute their coordinates into the line's equation. Think of the line as a boundary: one point gives a positive value, the other gives a negative value. You'll substitute each point into $3x - 5y + a$ and set up an inequality that forces these results to have opposite signs—that constraint will pin down the range of possible values for $a$.

The key insight is that two points lie on opposite sides of a line if and only if they produce opposite signs when you substitute their coordinates into the line's equation. Think of the line as a boundary: one point gives a positive value, the other gives a negative value. You'll substitute each point into $3x - 5y + a$ and set up an inequality that forces these results to have opposite signs—that constraint will pin down the range of possible values for $a$.