5. Solve for   x   and check.

The first step is to clear denominators. Multiplying both sides by the common denominator of   4(2x - 3)   will result in cross multiplying

4(1) = x(2x - 3)

4 = 2x2 - 3x

This is a quadratic, so we transpose everything to one side leaving a zero on the other. In this case, the square term is positive on the right, so it would be a good to transpose everything to the right to side.

0 = 2x2 - 3x - 4

This does not factor so let's use the quadratic formula.

There are two solutions. We can check them both at once. Copy down the original equation except that instead of copying down an x,copy down the solution in partnthese.

It doesn't look too encouraging. It will take a miracle for this to come out right. Let's just apply our techniques. Find common denominators for the fractions on the bottom on the left,and invert the bottom and multiply on the right.

We can invert and multiply at the same time as we subtract the 6 from the 3 on the left.

It's still not looking very good. But we really should rationalize the denominator on the left. To rationalize the denominator when there are two terms and at least one term involves radicals, multiply the top and bottom by the conjugate. We get the conjugate by changing the sign between the two terms.

Whether the radicals on the left are positive or negative, the left side will come out to be.

and amazingly it checks. The checks to these equations provide an amazing amount of fun.