Needgmat wrote:Is |xy|>x^2y^2 ?
1) 0 < x^2 < 1/4
2) 0 < y^2 < 1/9
\[\,{\left( {xy} \right)^2}\,\,\mathop < \limits^? \,\,\,\left| {xy} \right|\]
\[\left( 1 \right){\kern 1pt} \,\,\,0 < {x^2} < \frac{1}{4}\,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {x\,;\,y} \right) = \left( {\frac{1}{3}\,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\,\,Take\,\,\left( {x\,;\,y} \right) = \left( {\frac{1}{3}\,\,;\,\,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right){\kern 1pt} \,\,\,0 < {y^2} < \frac{1}{9}\,\,\,\,\left\{ \begin{gathered}
\,\,Take\,\,\left( {x\,;\,y} \right) = \left( {0\,\,;\,\,\frac{1}{4}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\,\,Take\,\,\left( {x\,;\,y} \right) = \left( {1\,\,;\,\,\frac{1}{4}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\,\,\]
\[\left( {1 + 2} \right){\kern 1pt} \,\,\,\]
\[\left. \begin{gathered}
0 < {x^2} < \frac{1}{4}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered}
\,\mathop \Rightarrow \limits^{\sqrt {} } \,\,\,\,0 < \left| x \right| < \frac{1}{2} \hfill \\
\mathop \Rightarrow \limits^{0\, < \,\,\left| x \right|\, < \,\,1} \,\,0 < {\left| x \right|^{\,2}} < \left| x \right|\,\,\,\,\,\, \hfill \\
\end{gathered} \right.\mathop \Rightarrow \limits^{{{\left| x \right|}^{\,2}} = \,\,{x^2}} \,\,\,\,\,0 < {x^2} < \left| x \right|\,\,\,\,\,\,\,\,\,\, \hfill \\
0 < {y^2} < \frac{1}{9}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered}
\,\mathop \Rightarrow \limits^{\sqrt {} } \,\,\,\,0 < \left| y \right| < \frac{1}{3} \hfill \\
\mathop \Rightarrow \limits^{0\, < \,\,\left| y \right|\, < \,\,1} \,\,0 < {\left| y \right|^{\,2}} < \left| y \right|\,\,\,\,\,\, \hfill \\
\end{gathered} \right.\mathop \Rightarrow \limits^{{{\left| y \right|}^{\,2}} = \,\,{y^2}} \,\,\,\,\,0 < {y^2} < \left| y \right| \hfill \\
\end{gathered} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,0 < {\left( {xy} \right)^2} < \left| {xy} \right|\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \]
The above follows the notations and rationale taught in the GMATH method.
