What is 5/(x + 2) - 8/(x + 4) in its simplest form?

Question

What is 5/(x + 2) - 8/(x + 4) in its simplest form?

Answer 1

Lets see, assuming x is not equal to -2 and -4 which is a valid assumption otherwise the original equation would be undefined, then:

you can multiply the top and bottom of 5/(x+2)  by (x+4)  and the top and bottom of -8/(x+4) by (x+2)

this gives:

5(x+4)/[(x+2)(x+4)] - 8(x+2)/[(x+2)(x+4)]

they have the same denominator and so we can combine them now. This is equal to

[5(x+4) - 8(x+2)]/[(x+2)(x+4)]

which is

(5x + 20 - 8x - 16)/[(x+2)(x+4)]

which simplifies to

(-3x + 4)/[[(x+2)(x+4)]

to test this, lets try x = 0

the original comes out to 5/2 - 8/4 = 1/2

the simplified version we get 4/8 = 1/2

and if you try it out with other numbers it should work too

I can't really see much more simplification

Answer 2

=> 8(x+2) = 5(x+4)

=> 8x + 16 = 5x + 20

=> 3x = 4

=> x = 4/3

=> x = 1 1/3