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

Sep 25, 2015

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 $\frac{2 x + 1}{x - 9}$,

the sign MIGHT and does change at $x = - \frac{1}{2}$ (the solution to $2 x + 1 = 0$ and

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

In the graph of $y = \frac{2 x + 1}{x - 9}$ 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 $x = - \frac{1}{2}$, y is positive. Then at $x = - \frac{1}{2}$, $y$ 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.