Question

How do you find the critical points for the inequality `(2x+1)/(x-9)>=0`?

Answer 1

See the explanation.

Explanation:

I think the question wants the key numbers or the partition numbers.

These are the values of `x` at which the sign of the expression MIGHT change.

The sign of a rational function MIGHT change when either the numerator is `0` or when the denominator is `0` (where the expression is not defined.)

For the expression `(2x+1)/(x-9)`,

the sign MIGHT and does change at `x=-1/2` (the solution to `2x+1=0` and

at `x=9` (the solution to `x-9=0`).

In the graph of `y=(2x+1)/(x-9)` below, you can see where the expression changes sign:

graph{y=(2x+1)/(x-9) [-25.9, 39.05, -23.36, 9.1]}

To the left of `x=-1/2`, y is positive. Then at `x=-1/2`, `y` changes from positive to negative. (You can use a mouse to scroll in and to drag the graph around.)
`y` stays negative until we get to `x=9` where `y` suddenly becomes very,very positive.