Q:  Solve for x:  5/(x+4) = 4 + 3/(x-2)

A:  I want to clear the denominators on this problem to make it easier to solve.  The two denominators are:

(x + 4) and (x – 2)

I am going to multiply the whole equation by (x + 4)(x – 2) like so:

(x + 4)(x – 2) [ 5/(x+4) = 4 + 3/(x-2) ]

Distribute through:

5(x + 4)(x – 2)/(x+4) = 4(x + 4)(x – 2) + 3(x + 4)(x – 2)/(x-2)

Now, cancel things out:

5(x + 4)(x – 2)/(x+4) = 4(x + 4)(x – 2) + 3(x + 4)(x – 2)/(x-2)

5(x – 2) = 4(x + 4)(x – 2) + 3(x + 4)

5x – 10 = 4(x² – 2x + 4x – 8 ) + 3x + 12

5x – 10 = 4x² + 8x – 32 + 3x + 12

5x – 10 = 4x² + 11x – 20

0 = 4x² + 6x – 10

…. Now solve using the quadratic equation…

x = [-b ± √(b² – 4ac)] / (2a)

x = [-6 ± √(6² – 4*4*-10)] / (2*4)

x = [-6 ± √(36 + 160)] / (8)

x = [-6 ± √(196)] / (8)

x = [-6 ± 14] / (8)

x = [-6 + 14] / (8)  or x = [-6 – 14] / (8)

x = [8] / (8)  or x = [-20] / (8)

x = 1  or x = -5/2