Question

How do you solve `1/(x+4)-2=(3x-2)/(x+4)`?

Answer 1

`x=-1`

Explanation:

`1/(x+4) -2 = (3x-2)/(x+4)`

Get a common denominator by multiplying the 2nd term by `(x+4)/(x+4)`

`1/(x+4)-(2 * (x+4))/((x+4))= (3x-2)/(x+4)`

`1/(x+4)- ((2x+8))/(x+4)=(3x-2)/(x+4)`

Compare to a simple equation like `1/5 + 2/5 =x/5`
Because there is a common denominator, this can be solve with the numerators only `1 +2 =x`

Similarly, solve the equation using only the numerators.

`1-(2x+8)=3x-2`

Distribute the negative.

`1-2x-8=3x-2`

`-2x-7=color(white)(aa)3x-2`
`+2x+2=+2x+2`

`-5=5x`

`-5/5 =(5x)/5`

`-1=x`

When solving rational equations, it is important to check for extraneous solutions (solutions that "don't work" or result in division by zero).

`1/(-1+4)-2=frac{3(-1)-2}{-1+4}color(white)(aaa)`True