Question

How do you solve $ \dfrac{1}{2} + \dfrac{2}{x} = \dfrac{1}{x} $ ?

Answer 1

Hint: In this question, we need to solve $ \dfrac{1}{2} + \dfrac{2}{x} = \dfrac{1}{x} $ . Here, we will take LCM, then cross-multiply the terms, by which we will get an equation. Thus, we will equate the equation for $ 0 $ , by which we will get the required value of $ x $ .

Complete step-by-step answer:
Here, we need to solve the given equation $ \dfrac{1}{2} + \dfrac{2}{x} = \dfrac{1}{x} $ .
Now, let us take LCM in the LHS,
The LCM of the terms are $ 2x $ ,
Therefore, $ \dfrac{1}{2} \times \dfrac{x}{x} + \dfrac{2}{x} \times \dfrac{2}{2} = \dfrac{1}{x} $
 $ \dfrac{x}{{2x}} + \dfrac{4}{{2x}} = \dfrac{1}{x} $
 $ \dfrac{{x + 4}}{{2x}} = \dfrac{1}{x} $
By cross multiplication, we get,
 $ x\left( {x + 4} \right) = 2x\left( x \right) $
 $ {x^2} + 4x = 2{x^2} $
Now, by rearranging the terms, we get,
 $ 2{x^2} - {x^2} - 4x = 0 $
 $ {x^2} - 4x = 0 $
Thus, taking out the $ x $ as common, we have,
 $ x\left( {x - 4} \right) = 0 $
By equating the terms to zero,
 $ x = 0 $ and $ x - 4 = 0 $
Therefore, $ x = 4 $
Hence by solving the given equation, we have the value of $ x = 4 $
So, the correct answer is “$ x = 4 $”.

Note: In this question it is important to note here that when we are facing these types of questions, the first basic thing we need to do is, simplify each side of the equation by removing parentheses and combining like terms. Then, isolate the variable term on one side of the equation by using addition or subtraction. And, use multiplication or division to solve for the variable. Suppose, if we are applying an operation to one side of the equation then we must apply to the other side of the equation also. If we didn’t apply then it would be wrong. For example: squaring on both sides, etc. by maintaining this rule, we can change the way the terms of an equation are written without changing their relation to each other. Fractions may be removed by multiplying each side of the equation by the common denominator.