Answer

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Hint: In this question first simplify the L.H.S part of the equation and later on apply the property of cross multiplication, so use these concepts to reach the solution of the question.

Complete step-by-step answer:

Given equation is

$\dfrac{{4x}}{{x - 2}} - \dfrac{{3x}}{{x - 1}} = 7\dfrac{1}{2}$

As we know $7\dfrac{1}{2}$is also written as $\dfrac{{\left( {7 \times 2} \right) + 1}}{2} = \dfrac{{15}}{2}$, substitute this value in above equation we have,

$\dfrac{{4x}}{{x - 2}} - \dfrac{{3x}}{{x - 1}} = \dfrac{{15}}{2}$

Now take L.C.M of the above equation we have,

$\dfrac{{4x\left( {x - 1} \right) - 3x\left( {x - 2} \right)}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} = \dfrac{{15}}{2}$

Now simplify the numerator and denominator of the L.H.S we have

$ \Rightarrow \dfrac{{4{x^2} - 4x - 3{x^2} + 6x}}{{{x^2} - x - 2x + 2}} = \dfrac{{15}}{2}$

$ \Rightarrow \dfrac{{{x^2} + 2x}}{{{x^2} - 3x + 2}} = \dfrac{{15}}{2}$

Now apply cross multiply we have

$ \Rightarrow 2\left( {{x^2} + 2x} \right) = 15\left( {{x^2} - 3x + 2} \right)$

Now simplify the above equation we have,

$

\Rightarrow 2{x^2} + 4x = 15{x^2} - 45x + 30 \\

\Rightarrow 15{x^2} - 2{x^2} - 45x - 4x + 30 = 0 \\

\Rightarrow 13{x^2} - 49x + 30 = 0 \\

$

Now divide by 13 in the above equation we have,

$ \Rightarrow {x^2} - \dfrac{{49}}{{13}}x + \dfrac{{30}}{{13}} = 0$

Now factorize the above equation we have,

$ \Rightarrow {x^2} - 3x - \dfrac{{10}}{{13}}x + \dfrac{{30}}{{13}} = 0$

$

\Rightarrow x\left( {x - 3} \right) - \dfrac{{10}}{{13}}\left( {x - 3} \right) = 0 \\

\Rightarrow \left( {x - 3} \right)\left( {x - \dfrac{{10}}{{13}}} \right) = 0 \\

\Rightarrow \left( {x - 3} \right) = 0,{\text{ }}\left( {x - \dfrac{{10}}{{13}}} \right) = 0 \\

\Rightarrow x = 3,{\text{ }}\dfrac{{10}}{{13}} \\

$

So, this is the required solution of the question.

Hence, option (b) is correct.

Note: In such types of questions simplification is the key, so simplify the above equation as above doing simplification don’t make unnecessary mistakes it will lead us to wrong answer so be careful while doing addition, subtraction, division and multiplication then apply cross multiply and again simplify then factorize the equation we will get the required solution of the x.

Complete step-by-step answer:

Given equation is

$\dfrac{{4x}}{{x - 2}} - \dfrac{{3x}}{{x - 1}} = 7\dfrac{1}{2}$

As we know $7\dfrac{1}{2}$is also written as $\dfrac{{\left( {7 \times 2} \right) + 1}}{2} = \dfrac{{15}}{2}$, substitute this value in above equation we have,

$\dfrac{{4x}}{{x - 2}} - \dfrac{{3x}}{{x - 1}} = \dfrac{{15}}{2}$

Now take L.C.M of the above equation we have,

$\dfrac{{4x\left( {x - 1} \right) - 3x\left( {x - 2} \right)}}{{\left( {x - 2} \right)\left( {x - 1} \right)}} = \dfrac{{15}}{2}$

Now simplify the numerator and denominator of the L.H.S we have

$ \Rightarrow \dfrac{{4{x^2} - 4x - 3{x^2} + 6x}}{{{x^2} - x - 2x + 2}} = \dfrac{{15}}{2}$

$ \Rightarrow \dfrac{{{x^2} + 2x}}{{{x^2} - 3x + 2}} = \dfrac{{15}}{2}$

Now apply cross multiply we have

$ \Rightarrow 2\left( {{x^2} + 2x} \right) = 15\left( {{x^2} - 3x + 2} \right)$

Now simplify the above equation we have,

$

\Rightarrow 2{x^2} + 4x = 15{x^2} - 45x + 30 \\

\Rightarrow 15{x^2} - 2{x^2} - 45x - 4x + 30 = 0 \\

\Rightarrow 13{x^2} - 49x + 30 = 0 \\

$

Now divide by 13 in the above equation we have,

$ \Rightarrow {x^2} - \dfrac{{49}}{{13}}x + \dfrac{{30}}{{13}} = 0$

Now factorize the above equation we have,

$ \Rightarrow {x^2} - 3x - \dfrac{{10}}{{13}}x + \dfrac{{30}}{{13}} = 0$

$

\Rightarrow x\left( {x - 3} \right) - \dfrac{{10}}{{13}}\left( {x - 3} \right) = 0 \\

\Rightarrow \left( {x - 3} \right)\left( {x - \dfrac{{10}}{{13}}} \right) = 0 \\

\Rightarrow \left( {x - 3} \right) = 0,{\text{ }}\left( {x - \dfrac{{10}}{{13}}} \right) = 0 \\

\Rightarrow x = 3,{\text{ }}\dfrac{{10}}{{13}} \\

$

So, this is the required solution of the question.

Hence, option (b) is correct.

Note: In such types of questions simplification is the key, so simplify the above equation as above doing simplification don’t make unnecessary mistakes it will lead us to wrong answer so be careful while doing addition, subtraction, division and multiplication then apply cross multiply and again simplify then factorize the equation we will get the required solution of the x.

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