# math

posted by .

how do you solve this inequality
3x-2 < x+4/x-2

• math -

You must mean:
3x-2 < (x+4)/(x-2) , so clearly x ≠ 2
(x-2)(3x-2) < x+4
3x^2 - 8x + 4 - x - 4 < 0
3x^2 - 9x < 0
3x(x-3) < 0

So the "critical values" are x=0 , x = 2, and x=3

test for a value of x<0
let x=-5 in original
-17 < -1/-7
-17 < 1/7 ? YES

test for a value between 0 and 2
let x=1
1 < 5/-1
1 < -5 ? NO

test for a value between 2 and 3
let x = 2.5
5.5 < 6.5/.5
5.5 < 13 ? , YES

test for a value x>3
let x=5
13 < 9/3
13 < 3 ? , NO

so we have x ≤ 0 OR 2 < x ≤ 3 , notice that x=2 in not included

To see that this is correct
go to my favourite graphing program
http://rechneronline.de/function-graphs/
and enter
3x-2 - (x+4)/(x-2)

in the window for "first graph"
change the "Range y-axis from" entries to -200 to 200

you will see the curve below the x-axis from -infinitity to 0
above the x-axis from 0 to the asymptote of 2
below the x-axis from 2 to 3, and
above the x-axis for x>3

• math -

thanks so much :)

• math -

what if all the values you put in do not work. how would you write the intervals.?

• math -

If the inequality can not be satisfied for any real value of x, then you just have to say that.