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Question 1019700:
SOLVE THE INEQUALITY ALGEBRAICALLY:
2/X+1 >= 1/X-2
I rearranged this to: 5-x/(x-2)(x+1)<=0
The teacher's solution/answer states (-1,2) U (6,infinity)
I got 5 instead of 6. ???
HELP PLEASE =)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i graphed both equation and this is what i got.
so, it appears that you are right.
it should be greater than or equal to 5.
the interval would show as [5,infinity).
since it was greater than or equal, rather than just greater than, x = 5 is a good solution as well as everything to the right of x = 5.
the first graph shows each equation individually.
the second graph shows the first expression minus the second expression.
so the graphical solution shows you're right.
i had a little more trouble with the algebraic solution, but i think i finally figured it out.
these problems can be tricky and it always takes me more than one try to get them.
start with 2/(x+1) >= 1/(x-2)
subtract 1/(x+2) from both sides of the equation to get:
2/(x+1) - (/x-2) >= 0
put everything under a common denominator to get:
(2*(x-2) - (x+1) / ((x+1)*(x-2)) >= 0
simplify to get (2x - 4 - x - 1) / ((x+1)*(x-2)) >= 0
simplify further to get (x-5) / ((x+1)*(x-2)) >= 0
since (x-5) = 0 when x = 5, it's clear that x = 5 has to be in the solution set.
it is also clear that x cannot be -1 or 2 because then you would have a division by 0.
so there are 3 points to graph and find the sign of the equation between.
they are x = -1, x = 2, and x = 5.
there are vertical asymptotes at x = -1 and x = 2
all that's left is to figure out whether the equation is positive or negative in between the intervals shown.
when x = -2 (smaller than -1), then the equaation is equal to -1.75 which is negative.
when x = 7 (greater than 5), then the equation is equal to .05 which is positive.
when x = 0 (between -1 and 2), then the equation is equal to 2.5 which is positive.
i think i did the math correctly.
the algebra confirms the graph and vice versa so the solution looks good.
i've seen several instances where the answer sheet is wrong, so it's not to be unexpected.
just the fact that the equation was equal to 0 when x = 5 proves you were right.
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