4. Solve for   x   and check.

To clear denominators we multiply by the smallest common denominator which in this case is   x - 3.

The large parentheses on the left side are there because we had the sum of fractions together before we multiplied. However, these parentheses tell us how to proceed. We multiply the factors in front of the parentheses by both terms inside the parentheses.

This will clear out all of the denominators. Assemble the surviving factors.

2x - 3x(x-3)= 6

Remove parentheses,

2x - 3x2 + 9x = 6

and combine like terms.

-3x2 + 11x = 6

At this point we see that we have a quadratic. Transpose all of the terms to one side and leave a 0 on the other. In this case, the   x2   term is negative on the left, so it would be better to transpose all of the terms to the right leaving a 0 on the left.

0 = 3x2 - 11x + 6

Factor

(3x - 2)(x - 3) = 0

and set the factors = 0.

Which gives us two solutions,   x = 3,   and   x = 2/3.

Check: Let   x = 3.   Copy the original equation except that where there is an   x  , substitute a  3.

and we get a zero denominator when we check. The convention is that if a solution gives you zero denominator when you check, then the solution doesn't check.

What went wrong? The way the solution process works is that if

then the eqution on the next line is true, and if that is true, then the equaton on the next line is true, and so forth until we can conclude that if

then   x = 3   or   x = 2/3.   However, we would like to conclude that if   x = 3   or   x = 2/3, then

which is obtained by switching the hypothesis and the conclusion in the "if - then" statement. If you switch the hypothesis and conclusion in a true "if - then" statement, the result is not necessarily true. It it will be true if all of the steps in the derivation are reversible, but our first step, to multoply both sides by a common denominator, is not necessarily reversible. It will be reversible so long at the thing by which we muultplied both sides is not 0, but if   x = 3,   then it would be, and it is not surprising that   x = 3   does not check. Such a solution is called an extraneous solution.

There are two ways to get a solution which does not check. The most common way is to make a mistuake in the solution. If you don't make a mistake in the solution, the only way to get a solution which does not check is if it will give you a zero denominator when you do the check. How will you know if you will get a zero denominator when you do the check? You will have to do the check.

There is another way to solve this equation which will eliminate this problem

transpose the fraction to the left and the other term to the right.

Note that we have common denominators on the left.

This reduces.

Which gives us

2 = 3x

which gives us only the solution of   x = 2/3.

What about this other solution?

Let   x = 2/3.

Copy down the original equation except that when you see an x, copy down the solution in parentheses instead.

and it checks.