SOLUTION: I have been working on this Rational inequality since Saturday morning and am still miffed. Problem: (x/x-2)>=5 I put the expression into standard from by (x)/(x-2)-5>=0 I u

Question 31488This question is from textbook College Algebra
: I have been working on this Rational inequality since Saturday morning and am still miffed.
Problem: (x/x-2)>=5
I put the expression into standard from by (x)/(x-2)-5>=0
I used the LCD to factor and put into one single fraction for(x)-5(x-2)/(x-2)>=0.
Which gave me: (X-5x-10)/(x-2)>=0
After combining terms I had: (4x-10)/(x-2)>=0
Which I then set both numerator and demonitor to be solved for >=0 and arrivd at:
x>=10/4 or 5/2
and x-2=0 which is (+2-2)>=0 which is Undefined and not a solution since a zero in the denominnator can't be a solution.
So (2,5/2) became the boundary points to test for true and fasle statements.
This is where I get confused.
When I plugged this expression into my y= screen on my TI-83 Plus calculator it gave me a vertical asympotote where the solutions I came up with is located on the x-axis.
When I algebraically used test values for the three intervals of (-00,2) (2,5/2) (5/2,00) I was even more confused because, when I tested using (-2) for the first interval of (00,2) the result was (-4) a value that was not >=5 and for the next test interval of (2,5/2) and used (2.25) as a test value I still get a value less than 5.
And for the last test inteval of (5/2,00) I used (3) I still get a denominator (3-2) that is less than 5. So none of the solutions I came up with are true statements.
None of these test values would satisfy the inequality statement of being >= to 5.

So the only thing I could figure out was that the vertical asymptote must mean some sort of sign change that takes place, but whee in the solution process this takes place is a mystery to me, as I hunted til the cows did not come home trying to find this answer.
At this rate I will never get this take home quiz finished by Tuesday March 28th.
Your help in helping me with this task is greatly appreciated.
My God I have spent some 20 hours since Sat morning and its now nearly midnight Sunday.
Sincerely,
Robert S. This question is from textbook College Algebra

Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website!
SORRY TO NOTE YOUR PLIGHT.THOUGH YOU HAVE PROCEEDED IN A PROPER WAY,I SHALL GIVE YOU AN ALTERNATE ROUTE.SEE IF THAT SUITS YOU.
Problem: (x/x-2)>=5 ..YOU TRANSFERRED 5 TO LHS WHICH IS THE PROPER WAY IN INEQUALITIES`AS YOU DONOT KNOW ABOUT NATURE OF X AND ITS FUNCTIONS.BUT THERE IS ANOTHER WAY TO GO I AM JUST SHOWING IT TO YOU AS AN ILLUSTRTION
WE DONT KNOW NATURE OF X-2...SO....
CASE 1...X-2=0...NOT ALLWED AS THE FUNCTION IS NOT DEFINED THEN.
CASE.2. .....LET X-2 BE POSITIVE OR X>2..THEN
X>5(X-2)
X-5X>-10
-4X>-10
4X<10.....AS MULTIPLICATION WITH NEGATIVE NUMBER WILL REVERSE THE INEQUALITY.
X<10/4=2.5...NOW CHECK WITH OUR ORIGINAL ASSUMPTION THAT X>2..IT IS COMPATABLE...
HENCE FOR THIS CASE THE SOLUTION IS X>2 BUT <2.5...THAT IS X IS BETWEN 2 AND 2.5
CASE.3.....LET X-2 BE NEGATIVE...OR....X<2
X<5(X-2)...AS MULTIPLICATION WITH NEGATIVE NUMBER WILL REVERSE THE INEQUALITY.
X-5X<-10
-4X<-10
4X>10......AS MULTIPLICATION WITH NEGATIVE NUMBER WILL REVERSE THE INEQUALITY.
X>10/4=2.5..NOW CHECK WITH OUR ORIGINAL ASSUMPTION THAT X<2..IT IS NOT COMPATABLE...
HENCE FOR THIS CASE THERE IS NO SOLUTION
HENCE THE TOTAL SOLUTION SET IS
X LIES BETWEEN 2 AND 2.5
OK GOT IT .IF STILL IN DOUBT OR YOU WANT TO LEARN MORE PLEASE COME BACK.
RELATED QUESTIONS
I am still very confused on how to simplify rational expressions: This is the problem I... (answered by richwmiller,josmiceli)

I have been working on this all day. could someone explain this problem to me i am... (answered by Earlsdon)

Good Saturday Morning! I am working on some problems this morning.... I need to find... (answered by Fombitz)

I am so bad at story problems, I can not seem to figure out which equations to... (answered by ankor@dixie-net.com)

I am working on dividing a rational expression with another rational expression and I am... (answered by josgarithmetic,MathTherapy)

Ok, I have been working on these two since 4 30 this morning and still can't figure them... (answered by stanbon,Fombitz)

We are currently working on rational expressions in my Intermediate Algebra class and I... (answered by rothauserc)

Good morning, I have been working on this problem..F(x)=sq rt of 2x-x/2 and x=t. I have... (answered by stanbon)

If the cos x is -(1/2) and the x is in the 2nd quadrant, find the cos 2x. (no calculator) (answered by lwsshak3)