Bring all the equations to the LHS.

\(\frac{(x-1)}{(x-3)}-\frac{(x-2)}{(x-4)}>0\)

Take the LCM, then we get,

\(\frac{x^2-5x+4-x^2+5x-6}{(x-3)(x-4)}>0\)

Now, x = 3 or x = 4.
Plotting this in wavy curve, we get.
(-∞, 3) to (4,+∞)

So a<x<b can be written as 3<x<4
b-a = 4-3 = 1

D is the answer.

Let´s see a "non-standard" (but absolutely rigorous) approach, very interesting for high-level candidates:

\({{x - 1} \over {x - 3}} > {{x - 2} \over {x - 4}}\,\,\,\left( * \right)\,\,\,\, \Leftrightarrow \,\,\,\,\,a < x < b\)

\(? = b - a\)


\(\left( * \right)\,\,\,\,\,\mathop \Leftrightarrow \limits^{ \cdot \left( {x - 3} \right)\left( {x - 4} \right)} \,\,\,\,\left\{ \matrix{
\,\left( {x - 1} \right)\left( {x - 4} \right) > \left( {x - 2} \right)\left( {x - 3} \right)\,\,\,{\rm{if}}\,\,\,\,\left( {x - 3} \right)\left( {x - 4} \right) > 0\,\,\,\left( {\rm{I}} \right) \hfill \cr
\left( {x - 1} \right)\left( {x - 4} \right) < \left( {x - 2} \right)\left( {x - 3} \right)\,\,\,{\rm{if}}\,\,\,\,\left( {x - 3} \right)\left( {x - 4} \right) < 0\,\,\,\left( {{\rm{II}}} \right) \hfill \cr} \right.\)

\(\left( {\rm{I}} \right)\,\,\,\,\left( {x - 1} \right)\left( {x - 4} \right) > \left( {x - 2} \right)\left( {x - 3} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\,{x^2} - 5x + 4 > {x^2} - 5x + 6\,\,\,{\rm{impossible!}}\)


\(\left( {{\rm{II}}} \right)\,\,\,\,\left( {x - 1} \right)\left( {x - 4} \right) < \left( {x - 2} \right)\left( {x - 3} \right)\,\,\,\,\, \Leftrightarrow \,\,\,\,{x^2} - 5x + 4 < {x^2} - 5x + 6\,\,\,{\rm{always}}\,\,\left[ {\,{\rm{for}}\,\,\left( {x - 3} \right)\left( {x - 4} \right) < 0\,} \right]\,\,!\,{\rm{!}}\)

\(\left( {x - 3} \right)\left( {x - 4} \right) < 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,3 < x < 4\)


\(? = 4 - 3 = 1\)


The correct answer is (D).


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

Regards,
Fabio.
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Classical Approach:

\({{x - 1} \over {x - 3}} > {{x - 2} \over {x - 4}}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{{x - 1} \over {x - 3}} - {{x - 2} \over {x - 4}} > 0\,\,\,\,\, \Leftrightarrow \,\,\,\,{{\left( {x - 1} \right)\left( {x - 4} \right)} \over {\left( {x - 3} \right)\left( {x - 4} \right)}} - {{\left( {x - 2} \right)\left( {x - 3} \right)} \over {\left( {x - 4} \right)\left( {x - 3} \right)}} > 0\,\,\)

\(\,\, \Leftrightarrow \,\,\,\,{{ - 2} \over {\left( {x - 3} \right)\left( {x - 4} \right)}} > 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\left( {x - 3} \right)\left( {x - 4} \right) < 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,3 < x < 4\)

Regards,
Fabio.
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