vaibhav101 wrote:if a < y < z < b , is |y-a| less than |y-b| ?
1) |z-a| < |z-b|
2) |y-a| < |z-b|
\[a < y < z < b\,\,\,\,\left( * \right)\]
\[\left| {y - a} \right|\,\,\,\,\mathop < \limits^? \,\,\,\left| {y - b} \right|\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,y - a\,\,\,\,\mathop < \limits^? \,\,\,b - y\,\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{y\,\,\,\,\mathop < \limits^? \,\,\,\frac{{a + b}}{2}}\]
\[\left( 1 \right)\,\,\,\left. \begin{gathered}
{\text{dist}}\left( {z,a} \right) < {\text{dist}}\left( {z,b} \right)\,\,\, \hfill \\
a < y < z < b\,\,\,\left( * \right) \hfill \\
\end{gathered} \right\}\,\,\,\,\, \Rightarrow \,\,\,\left( {y < } \right)\,\,z < \frac{{a + b}}{2}\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( 2 \right)\,\,\left. \begin{gathered}
{\text{dist}}\left( {y,a} \right) < {\text{dist}}\left( {z,b} \right) \hfill \\
a < y < z < b\,\,\,\left( * \right) \hfill \\
\end{gathered} \right\}\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,y < \frac{{a + b}}{2}\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \]
\[\left( {**} \right)\,\,\,y \geqslant \frac{{a + b}}{2}\,\,\,\, \Rightarrow \,\,\,\,{\text{dist}}\left( {y,a} \right) \geqslant {\text{dist}}\left( {y,b} \right) > {\text{dist}}\left( {z,b} \right)\,\,\, \Rightarrow \,\,\,\,{\text{impossible}}\]
The answer is
__D__
The above follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
