Solution 2.2:3b

From Förberedande kurs i matematik 1

First, we move all the terms over to the left-hand side,

Then, we multiply the top and bottom of all three terms by appropriate factors so that they have the same common denominator, in the following way,

4x4x−72x−32x−3−12x−34x−74x−7−(2x−3)(4x−7)(2x−3)(4x−7)=0

and so that we can rewrite the left-hand side giving

We expand the numerator

and simplify

This equation is satisfied when the numerator is zero (provided the denominator is not also zero) and this happens when

10x−14=0

which gives x=75.

It can easily happen that we calculate incorrectly, so we check that the answer x=75 satisfies the equation,

LHS =457457−7−1257−3=multiply top and bottom by 5=457457−755−1257−355=4747−75−527−35=44−5−514−15=−4−(−5)=1=RHS.