Question

State true or False:
If $\dfrac{{150}}{{x - 5}} - \dfrac{{150}}{x} = 1$ , then $x = 30$ or $x = 25$
A) True
B) False

Answer 1

Hint:
It is given in the question that If $\dfrac{{150}}{{x - 5}} - \dfrac{{150}}{x} = 1$ ,then $x = 30$ or $x = 25$ is true or false.
Firstly, we have to take the L.C.M of the equation and then after by cross multiplying of the equation we get a quadratic equation.
Then, we will find the roots of the equation if the roots match with the roots that are given in the equation then the question is true otherwise false.

Complete step by step solution:
It is given in the question that If $\dfrac{{150}}{{x - 5}} - \dfrac{{150}}{x} = 1$ , then $x = 30$ or $x = 25$ is true or false.
 $\dfrac{{150}}{{x - 5}} - \dfrac{{150}}{x} = 1$
 $\dfrac{{150}}{{x - 5}} = 1 + \dfrac{{150}}{x}$
Now, take the L.C.M on right hand side
 $ \Rightarrow \dfrac{{150}}{{x - 5}} = \dfrac{{x + 150}}{x}$
Now, cross multiply
  $ \Rightarrow 150x = {x^2} + 150x - 5x - 750$
 $ \Rightarrow 150x - 150x = {x^2} - 5x - 750$
 \[ \Rightarrow {x^2} - 5x - 750 = 0\]
By using method of splitting of middle term we will solve the above equation,
 \[ \Rightarrow {x^2} - 30x + 25x - 750 = 0\]
 $ \Rightarrow x\left( {x - 30} \right) + 25\left( {x - 30} \right) = 0$
 \[ \Rightarrow \left( {x - 30} \right) = 0\] or $\left( {x + 25} \right) = 0$
 $ \Rightarrow x = 30$ or $x = - 25$
Thus, we find the roots of the equation $\dfrac{{150}}{{x - 5}} - \dfrac{{150}}{x} = 1$ as $x = 30$ or $x = - 25$. But it is given in the question that roots of the equation $\dfrac{{150}}{{x - 5}} - \dfrac{{150}}{x} = 1$ are $x = 30$ or $x = 25$. Both the statements contradict each other.
Hence the answer is false.

Note:
L.C.M: In number theory, the least common multiple, lowest common multiple of two integer a and b, usually denoted by lcm \[\left( {a,b} \right)\] is the smallest positive integer that is divisible by both a and b. Since division of integers by zero is undefined, this definition has meaning only if a and b are both different from zero.