Question

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

Answer 1

Hint: In order to multiply the above given rational equation having variable $ x $ ,use the cross-multiplication method by multiplying the numerator of LHS with the denominator of the RHS and similarly the denominator of the LHS with the numerator of the RHS. Simplify the equation by expanding the parenthesis by using distributive property and FOIL rule . Combine all the like terms to get the required result.

Complete step by step solution:
We are given a rational equation in variable $ x $
 $ \dfrac{{x + 1}}{{2x}} = \dfrac{{x - 1}}{{2x - 1}} $
Multiplication of the above equation will be a cross-multiplication of the terms.
In other words, the numerator of the left-hand side term will be multiplied with the denominator of the right-hand side term and similarly the denominator of LHS with the numerator of RHS.
 $ \left( {x + 1} \right)\left( {2x - 1} \right) = \left( {2x} \right)\left( {x - 1} \right) $
Simplifying the above equation and expanding the terms by resolving the parenthesis .For the right-hand side use distributive property of multiplication as $ A\left( {B + C} \right) = AB + AC $ and to expand the left-hand side binomial expression there is a order by which we can expand the expression easily. You can use the acronym FOIL which means First, Outside,inside,Last to remember the order of expanding .
We now have our equation as
 $
\Rightarrow x\left( {2x - 1} \right) + 1\left( {2x - 1} \right) = 2{x^2} - 2x \\
\Rightarrow 2{x^2} - x + 2x - 1 = 2{x^2} - 2x \;
  $
Now combining all the terms, and to do so, first transpose all the terms from left-hand side to right-hand side
 $
\Rightarrow 0 = 2{x^2} - 2{x^2} - 2x + x + 1 - 2x \\
\Rightarrow 0 = - 3x + 1 \;
  $
 $ - 3x + 1 = 0 $
Therefore the multiplied form of the given equation is $ - 3x + 1 = 0 $ .
So, the correct answer is “ $ - 3x + 1 = 0 $ ”.

Note: Alternatively to find the multiplied form the given equation , you can alternatively simply multiply both sides of the equation with $ \left( {2x} \right)\left( {2x - 1} \right) $ and simplify the equation to get the proper result.