(x+1) / (x-1) equal to or less than 2. Interval notation?

Question

(x+1) / (x-1) equal to or less than 2. Interval notation?

Answer 1

First, multiply the entire thing by (x - 1)

You get: (x + 1) <= 2 (x - 1)

Now, since you don't know if the inequality sign will shift directions because of the multiplication of (x - 1) (you don't know if x - 1 is positive or negative), replace the inequality sign with a blank.

Solve for x.

x + 1 __ 2x - 2

x __ 3

So you got a value of 3. Is this x >= 3 or x <= 3?

Let's take the value x = 2 for x <= 3 and x = 4 for x >= 3.

For x = 2:

(2 + 1) / (2 - 1) <= 2 ?

3 / 1 <= 2 (wrong!)

For x =  4:

(4 + 1) / (4 - 1) <= 2?

5 / 3 <= 2 (right!)

Therefore, the answer is x >= 3

In interval notation, this is [3, infinity)