Question

How do you solve for u in `4/(u+6)=6/(u+6)+2`?

Answer 1

`u=-7`

Explanation:

First, we need to put the requirements, that is `u!=-6` because then the denominator will be `0`, and make the equation undefined.

Then,

`4/(u+6)=6/(u+6)+2`

`-2/(u+6)=2`

`u+6=-1`

`u=-7`

Answer 2

A different approach: `u=-7`

Explanation:

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

`color(green)(4/(u+6)=6/(u+6)+2 color(white)("d")->color(white)("d")4/(u+6)=6/(u+6)+[2color(red)(xx1)])`

`color(green)( color(white)("ddddddddddddddddd")->color(white)("d")4/(u+6)=6/(u+6)+[2color(red)(xx(u+6)/(u+6))])`

`color(green)(color(white)("dDDDDddddddddddd")->color(white)("d")4/(u+6)=6/(u+6)+color(white)("d")(2(u+6))/(u+6)`

Now all the denominators (bottom numbers) are the same we can forget about them.

Or, as a purist would say: multiply all of both sides by `(u+6)`. This cancels out the denominators which is THE SAME THING!

`color(green)(4=6+2(u+6))`

`color(green)(4=6+2u+12)`

`color(green)(4=18+2u)`

Subtract 18 from both sides

`color(green)(2u=-14)`

Divide both sides by 2

`color(green)(u=-7)`

`color(blue)("Foot note: as "u" can only take on one value for this to work ")` `color(blue)("and this is not -6 then we do not need to state that "x!=-6)`