Quote:
Is |p + 2| - p > 5?
(1) |p| < 2
(2) p > p^2
\(\left| {p + 2} \right| - p = \left\{ \begin{gathered}
\left( {p + 2} \right) - p = 2\,\,\,\,\,if\,\,\,p \geqslant 2 \hfill \\
\left( {-p - 2} \right) - p = -2p - 2\,\,\,\,\,if\,\,\,p < 2 \hfill \\
\end{gathered} \right.\)
\(?\,\,\,\,\,:\,\,\,\,\,\,\left\{ \begin{gathered}
2\,\,\,\,\mathop > \limits^? \,\,\,5\,\,\,\,\,\,\,\,if\,\,\,p \geqslant 2\, \hfill \\
-2p - 2\,\,\,\,\mathop > \limits^? \,\,\,5\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,p\,\,\mathop < \limits^? - \frac{7}{2}\,\,\,\,\,\,\,if\,\,\,p < 2\, \hfill \\
\end{gathered} \right.\)
(1) -2 < p < 2 , hence we are dealing with the second expression in the last two possibilities. And NO, p is not less than -3.5 , for sure. Sufficient.
(2) \(p > {p^2}\,\,\, \Leftrightarrow \,\,\,p\left( {p - 1} \right) < 0\,\,\, \Leftrightarrow \,\,\,0 < p < 1\)
We are again dealing with the second expression in the last two possibilities. And NO, p is not less that -3.5, for sure. Sufficient.
The solution above follows the notations and rationale taught in the GMATH method.
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Fabio Skilnik ::
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