Posted by **ted** on Tuesday, July 24, 2012 at 8:07am.

how do you solve this inequality

3x-2 < x+4/x-2

please help i don't get it

- math -
**Reiny**, Tuesday, July 24, 2012 at 9:18am
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 -
**ted**, Tuesday, July 24, 2012 at 10:28am
thanks so much :)

- math -
**ted**, Tuesday, July 24, 2012 at 10:52am
what if all the values you put in do not work. how would you write the intervals.?

- math -
**Damon**, Tuesday, July 24, 2012 at 11:40am
If the inequality can not be satisfied for any real value of x, then you just have to say that.

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