I is true. According to the graph, we have
$\begin{array}{rcl}
\text{the $x$-intercept of $L$} & > & 0 \\
\dfrac{-15}{a} & > & 0 \\
a & < & 0
\end{array}$
and
$\begin{array}{rcl}
\text{the $y$-intercept of $L$} & > & 0 \\
\dfrac{-15}{b} & > & 0 \\
b & < & 0
\end{array}$
Note that the slope of $L$ is larger than the slope of the line passing through $(0,3)$ and $(3, 0)$. Then we have
$\begin{array}{rcl}
\dfrac{-a}{b} & > & \dfrac{3 – 0}{0 – 3} \\
\dfrac{-a}{b} & > & -1 \\
\dfrac{a}{b} & < & 1 \\
a & > & b
\end{array}$
II is true. According to the graph, we have
$\begin{array}{rcl}
\text{the $x$-intercept of $L$} & > & 5 \\
\dfrac{-15}{a} & > & 5 \\
-15 & < & 5a \\
a & > & -3
\end{array}$
III is not true. According to the graph, we have
$\begin{array}{rcl}
\text{the $y$-intercept of $L$} & < & 3 \\
\dfrac{-15}{b} & < & 3 \\
-15 & > & 3b \\
b & < & -5
\end{array}$
