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In the rectangular coordinate system, which of the above representatio

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GMATH Teacher
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In the rectangular coordinate system, which of the above representatio  [#permalink]

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13 Feb 2019, 15:19
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GMATH practice exercise (Quant Class 20)

In the rectangular coordinate system, which of the above representations better describes the points (x,y) such that y/(x-y) > 0?

(Alternative choices are presented in the figure.)

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: In the rectangular coordinate system, which of the above representatio  [#permalink]

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14 Feb 2019, 01:12
$$\frac{y}{(x-y)}>0$$

So we need either numerator&denominator both positives or both negatives, and $$x\neq{y}$$.
The latter constraint allows us to eliminate B as an option.

If y>0, we need x>y to have also a positive numerator -> eliminate C&E&D

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Re: In the rectangular coordinate system, which of the above representatio  [#permalink]

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14 Feb 2019, 05:29
fskilnik wrote:
GMATH practice exercise (Quant Class 20)

In the rectangular coordinate system, which of the above representations better describes the points (x,y) such that y/(x-y) > 0?

(Alternative choices are presented in the figure.)

$${\rm{?}}\,\,\,{\rm{:}}\,\,\,{\rm{best}}\,\,{\rm{graph}}\,\,{\rm{representation}}$$

$${y \over {x - y}}\,\,\mathop > \limits^{\left( * \right)} \,\,0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ \matrix{ \,y > 0\,\,\,{\rm{and}}\,\,\,x - y > 0 \hfill \cr \,\,\,{\rm{OR}} \hfill \cr \,y < 0\,\,\,{\rm{and}}\,\,\,x - y < 0 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ \matrix{ \,y > 0\,\,\,{\rm{AND}}\,\,\,\underline {x > y} \,\,\,\,\left( {{\rm{hence}}\,\,x > 0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\underline {{\rm{bottom}}\,\,{\rm{half}}} \,\,{\rm{of}}\,\,{\rm{1st}}\,{\rm{quadrant}} \hfill \cr \,\,\,{\rm{OR}}\,\,\,\,\, \hfill \cr \,y < 0\,\,\,{\rm{AND}}\,\,\,\underline {x < y} \,\,\,\,\left( {{\rm{hence}}\,\,x < 0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\underline {{\rm{upper}}\,\,{\rm{half}}} \,\,{\rm{of}}\,\,3{\rm{rd}}\,{\rm{quadrant}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{A}} \right)$$

$$\left( * \right)\,\,\,\left\{ \matrix{ \,y \ne 0\,\,\,\, \Rightarrow \,\,\,\,x{\rm{ - axis}}\,\,{\rm{excluded}} \hfill \cr x - y \ne 0\,\,\,\, \Rightarrow \,\,\,\,{\rm{line}}\,\,y = x\,\,{\rm{excluded}} \hfill \cr} \right.$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: In the rectangular coordinate system, which of the above representatio  [#permalink]

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14 Feb 2019, 21:41
So we have y/(x-y) >0
What we need to do is to guarantee the 2 scenarios:
Scenario 1: Numerator Positive and Denominator Positive

in other words: y>0 and x>y
This means 1st part of the solution is quadrant I and slope is more x than y (here we've got already the answer)

Scenario 2: Numerator Negative and Denominator Positive
in other words: y<0 and x<y
This means 2nd part of the solution is quadrant III and slope x is more than y

Re: In the rectangular coordinate system, which of the above representatio   [#permalink] 14 Feb 2019, 21:41
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