Data Sufficiency

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.

if a < y < z < b , is |y-a| less than |y-b| ?

1) |z-a| < |z-b|
2) |y-a| < |z-b|

|a-b| = the distance between a and b.

Since a < y < z < b, we get the following number line:
a..........y..........z..........b

Statement 1: |z-a| < |z-b|
Since the distance between y and a is less than the distance between z and a, we get:
|y-a| < |z-a|
Since the distance between z and b is less than the distance between y and b, we get:
|z-b| < |y-b|
Linking the inequalities above to the inequality in Statement 1, we get:
|y-a| < |z-a| < |z-b| < |y-b|
|y-a| < |y-b|
Thus, the answer to the question stem is YES.
SUFFICIENT.

Statement 2: |y-a| < |z-b|
Since the distance between z and b is less than the distance between y and b, we get:
|z-b| < |y-b|
Linking the inequality above to the inequality in Statement 2, we get:
|y-a| < |z-b| < |y-b|
|y-a| < |y-b|
Thus, the answer to the question stem is YES.
SUFFICIENT.

The correct answer is D.