4. Solve for x and check.

 

 

Again we have a rational equation. We first clear denominators. The smallest common denominator in this case is 4(2x - 3) Multiply both sides by the common denominator.

 

 

After cancelling we get

 

 

and we see that, in this case, clearing denominators amounts to cross multiplying. Remove parentheses

 

 

and we see that we have a quadratic equation. Transpose all terms to one side leaving a 0 on the other.

 

 

Unfortunately, as the reader can verify, this does not factor, so we need to use another method such as completing the square or the quadratic formula. Let's use the quadratic formula.

 

 

Simplify. Under the radical we will do the power and the product before we subtract.

 

 

or

 

 

Check: There are two answers to check here: one is obtained by adding the radical and the other is obtained by subtracting the radical. We can check them both at the same time. Copy down the original equation,

 

 

except that wherever there is an x, copy down the solution in parentheses.

 

 

This gives us compound fraction problems on both sides. Finish the subtraction in the denominator on the left. For this we will need common denominators.

 

 

 

We are now ready to invert and multiply

 

 

On the left, we need to rationalize the denominator.

 

 

 

 

and it checks

 

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