Question

How do you solve `4- \frac{1}{x + 2} = \frac{7}{x + 5}`?

Answer 1

`x=-3/2` or `x=-7/2`

Explanation:

`4-1/(x+2)=7/(x+5)`

`4=7/(x+5)+1/(x+2)`

`4=((7)(x+2)+(x+5))/((x+2)(x+5))`

`4(x+2)(x+5)=7(x+2)+(x+5)`

`4(x^2+7x+10)=7x+14+x+5`

`4x^2+28x+40=8x+19`

`4x^2+20x+21=0`

`(2x+3)(2x+7)=0`

`x=-3/2` or `x=-7/2`

Answer 2

`x=-7/2` or `x=-3/2`

Explanation:

Given
`color(white)("XXX")4-1/(x+2)=7/(x+5)`

Since I don't like working with fractions, let's multiply everything by `(x+2)(x+5)` giving:
`color(white)("XXX")4(x+2)(x+5)-(x+5)=7(x+2)`

then spend a few lines expanding and simplifying
`color(white)("XXX")4(x^2+7x+10)-x-5=7x+14`

`color(white)("XXX")4x^2+27x+35=7x+14`

`color(white)("XXX")4x^2+20x+21=0`

If you are really good, you might be able to factor the left side;
but I will just apply the quadratic formula to get
`color(white)("XXX")x=(-20+-sqrt(20^2-4 * 4 * 21))/(2 * 4)`

`color(white)("XXX"x)=(-20+-8)/8`

`color(white)("XXX")rArr x=(-28)/8=-7/2" or "x=(-12)/8=-3/2`