Question

How do you solve `6/(x+4 )+ 3/4 = (2x+1) / (3x+12)`?

Answer 1

`x = -104`

Explanation:

Your starting expression looks like this

`6/(x+4) + 3/4 = (2x+1)/(3x+12)`

Notice that you can use `3` as a common factor for the denominator of the fraction that's on the right-hand side of the equation

`3x + 12 = 3 * (x + 4)`

The equation can thus be written as

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

Next, you need to get rid of the denominators. To do that, you need to find the least common multiple of the expressions that act as denominators for your three fractions

`x+4" "`, `" "4" "`, and `" "3(x+4)`

Notice that you need to multiply the first one by `3` and by `4`, the second one by `3` and `(x+4)`, and the third one by `4` to get

`3 * 4 * (x+4)" "`

This will be your least common multiple. The equation will become

`6/(x+4) * 12/12 + 3/4 * (3(x+4))/(3(x+4)) = (2x+1)/(3(x+4)) * 4/4`

`(6 * 12)/(12(x+4)) + (3 * 3(x+4))/(12(x+4)) = ((2x+1) * 4)/(12(x+4))`

This is equivalent to

`72 + 9 * (x+4) = 4 * (2x + 1)`

Finally, expand the parantheses and isolate `x` on one side of the equation

`72 + 9x + 36 = 8x + 4`

`9x - 8x = 4 - 108`

`x = color(green)(-104)`

Do a quick check to make sure the calculations are correct

`6/(-104 + 4) + 3/4 = (2 * (-104) + 1)/(3 * (-104 + 4)`

`-0.06 + 0.75 = 0.69color(white)(x)color(green)(sqrt())`