How do I solve equations with unknowns in the denominators?

Suppose you have an equation: (2x+3)/(x-4) - (2x-8)/(2x+1) = 1 and you want to solve for x.

Remember that you can only add fractions if they share a common denominator. Also that you can multiply the numerator and the denominator by the same thing without changing the fraction.

The first thing we want to do is to create a common denominator on the left hand side. To do this, simply multiply the numerator and denominator of each fraction by the denominator of the other. This gives: [(2x+3)(2x+1)] / [(x-4)(2x+1)] - [(2x-8)(x-4)] / [(x-4)(2x+1)] = 1.

Because we now have a common denominator we can add the fractions: [(2x+3)(2x+1) - (2x-8)(x-4)] / (x-4)(2x+1) = 1.

We now multiply both sides of the equation by (x-4)(2x+1) to get: (2x+3)(2x+1) - (2x-8)(x-4) = (x-4)(2x+1).

To avoid confusion later on, I'm going to multiply (2x-8)(x-4) by -1 to get: (2x+3)(2x+1) + (2x-8)(4-x) = (x-4)(2x+1). I'm allowed to do this because of the minus sign, which effectively made the equation read (2x+3)(2x+1) + (-1)*(2x-8)(x-4) = ...

Now we can multiply out the brackets to get: 4x² + 6x + 2x + 3 - 2x² + 8x + 8x - 32 = 2x² + x - 8x -4.

Gathering like terms gives: 0x² + 31x - 25 = 0.

Solving this gives x = 25/31 = 0.81 to 2 decimal places.

The step by step solution to these kinds of equations is:

1: Multiply numerators and denominators to get a common denominator.

2: Add the fractions.

3: Multiply both sides by the denominator to eliminate the fraction.

4: Make the equation easier if you can. (This was where I removed the "-" sign).

5: Multiply out the brackets.

6: Gather like terms and solve.