6. Solve for x and check.
First, clear denominators. Multiply both sides by x + b.
Transpose all terms that involve x to one side and the terms that do not contain a factor of x to the other side.
Factor out the x
Divide by the coefficient of the unknown
c cannot be equal to one or else you will have a zero denominator in the solution. If c = 1, then the top of the fraction and the bottom have to be the same. That will happen only if a = b, in which case any value of x will be a solution. Otherwise, there will be no value of x which will work
Check. Copy down the original equation except where you see an x, copy down the solution in parentheses.
On the left we have a compound fraction. Add the fractions on both the top and bottom in order to prepare for inverting and multiplying. The common denominator is 1 - c.
After we add the fractions
we find some like terms that cancel.
We are now ready to invert and multiply.
Note that you can factor out a c from both terms in the left top.
After we cancel we get
Note that in the check, in addition to the requirement that c is not equal to 1, there is another occasion to get zero denominators in the check. If a = b there will be a zero denominator in the check. If a = b, then the fraction will always reduce to 1, and so any value of x would work if c were 1, but no value would work if c were not 1. In either case, there is not a unique solution, and this is manifested by the problems with zero denominators.