Question

How do you combine `(4a-2)/(3a+12)-(a-2)/(a+4)`?

Answer 1

See a solution process below:

Explanation:

To subtract or add fractions they must be over a common denominator. We can multiply the fraction on the right by the appropriate form of `1` to put these fractions over a common denominator:

`(4a - 2)/(3a + 12) - (3/3 xx (a - 2)/(a + 4)) =>`

`(4a - 2)/(3a + 12) - (3(a - 2))/(3(a + 4)) =>`

`(4a - 2)/(3a + 12) - (3a - 6)/(3a + 12)`

We can now subtract the numerators over the common denominator:

`((4a - 2) - (3a - 6))/(3a + 12) =>`

`(4a - 2 - 3a + 6)/(3a + 12) =>`

`(4a - 3a - 2 + 6)/(3a + 12) =>`

`((4 - 3)a + (-2 + 6))/(3a + 12) =>`

`(1a + 4)/(3a + 12) =>`

`(a + 4)/(3a + 12)`

We can now factor the numerator and cancel common terms:

`(a + 4)/(3(a + 4)) =>`

`color(red)(cancel(color(black)(a + 4)))/(3color(red)(cancel(color(black)((a + 4))))) =>`

`1/3`

Answer 2

This equals `1/3`, with restriction `a != -4`

Explanation:

We can factor the denominator of the left-most expression as `3(a + 4)`, thus we multiply the second fraction by `3`.

`=(4a - 2 - 3(a - 2))/(3a + 12)`

`=(4a - 2 - 3a + 6)/(3a + 12)`

`=(a + 4)/(3a + 12)`

`=(a + 4)/(3(a + 4))`

`=1/3`

But don't forget that `a != -4`.

Hopefully this helps!