If 7 < m < 11 and −2 < n < 5, then which of th

BTGmoderatorDC wrote:If 7 < m < 11 and −2 < n < 5, then which of the following expresses the possible values of mn ?

A. −14 < mn < 35
B. −14 < mn < 55
C. −22 < mn < 35
D. −22 < mn < 55
E. 5 < mn < 16

Source: Princeton Review

\[?\,\,:\,\,mn\,\,{\text{possibilities}}\]
\[\left\{ {n: - 2 < n < 5} \right\} = \left\{ {n: - 2 < n < 0} \right\} \cup \left\{ {n:0 \leqslant n < 5} \right\}\]
\[\left\{ \begin{gathered}
7 < m < 11 \hfill \\
0 \leqslant n < 5 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,0 \leqslant mn < 55\,\,\,\,\,\,\left( {\text{I}} \right)\]
\[\left\{ \begin{gathered}
7 < m < 11 \hfill \\
- 2 < n < 0 \hfill \\
\end{gathered} \right.\,\,\,\,\,\, \sim \,\,\,\,\,\,\,\left\{ \begin{gathered}
7 < m < 11 \hfill \\
0 < - n < 2 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,0 < - mn < 22\,\,\,\, \Rightarrow \,\,\,\, - 22 < mn < 0\,\,\,\,\,\,\left( {{\text{II}}} \right)\]
\[? = \left( {\text{I}} \right) \cup \left( {{\text{II}}} \right)\,\,\, \Rightarrow \,\,\, - 22 < mn < 55\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

BTGmoderatorDC wrote:If 7 < m < 11 and −2 < n < 5, then which of the following expresses the possible values of mn ?

A. −14 < mn < 35
B. −14 < mn < 55
C. −22 < mn < 35
D. −22 < mn < 55
E. 5 < mn < 16

To determine the lower limit and the upper limit of mn, calculate mn using every combination of endpoints (7 and 11 for m, -2 and 5 for n):
mn = (7)(-2) = -14.
mn = (7)(5) = 35.
mn = (11)(-2) = -22.
mn = (11)(5) = 55.

The LOWER limit is the LEAST of the results above: -22.
The UPPER limit is the GREATEST of the results above: 55.
Thus:
-22 < mn < 55.

The correct answer is D.