Question

How do you find the equations of the tangent lines to the curve `y= (x-1)/(x+1)` that are parallel to the line `x-2y = 2`?

Answer 1

Find the equations of the tangent lines to the curve `y= (x-1)/(x+1)` that are parallel to the line `x-2y = 2`.

There is a bit of algebra and arithmetic for this. Let's focus on the reasoning and the calculus.

One
A line parallel to `x-2y = 2` must have the same slope. The slope of this line is `1/2`. So we want the slope of the tangent line to be `1/2`

Two
How do we find the slope of the tangent line? -- The derivative. So, we want the derivative to be `1/2`.

What is the derivative of `y= (x-1)/(x+1)` ?

Use the quotient rule:

`y'= (x(x+1)-(x-1)*1)/(x+1)^2 = 2/(x+1)^2`

Three

Find `x` to make `y'=1/2`

`2/(x+1)^2 = 1/2` if and only if

`(x+1)^2 =4`

So `x+1 = +-2`

and `x=1, -3`

Four

Find the `y` values at `x=1` (`y=0`)and at `x=-3 (`y=2)#

Five

Find the equations of the lines:

through `(1,0)` with slope `m=1/2`

and through `(-3,2)` with slope `m=1/2`.

Answer 2

Have a look:

which gives yu two equations:
`y=1/2x+7/2`
and
`y=1/2x-1/2`