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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.
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