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

1 Answer
Sep 25, 2015

Answer:

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.