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GMATH practice exercise (Quant Class 11)

If \(\,{{x - 1} \over {x - 3}}\,>\,{{x - 2} \over {x - 4}}\,\) is equivalent to \(\,a < x < b\,\), the value of \(b-a\) is:

(A) 0.25
(B) 0.5
(C) 0.75
(D) 1
(E) 1.25
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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