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Question 28966: 3-2/3x_>1
Answer by sdmmadam@yahoo.com(530)   (Show Source):
You can put this solution on YOUR website! 3-2/3x_>1
3-(2/3)X(1/x) >= 1
(of course x is NOT ZERO here) (as division by zero is not defined)
3-1 >= (2/3)X(1/x)
2 >= (2/3)X(1/x)
1 >= (1/3x) ---(*)
Case 1: Let x >0
Then multiplying (*) by (x) >0
(multiplying by a positive quantity does not alter the inequality sign)
(And here greater than remains greater than.)
(1)X(x) >= (1/3x)X(x)
x >= 1/3
We have got the result x >= 1/3 for our consideration of x >0
Anything equal to 1/3 or to the right of 1/3 is of course positive.
And as something positive (that is to the right of zero but to the left of 1/3 cannot be >= to 1/3.
So the verdict is x >= 1/3
Case 2: Let x <0
Then multiplying (*) by (x) < 0
(multiplying by a negative quantity alters the inequality sign)(
And here greater than becomes less than.)
(1)X(x) =< (1/3x)X(x)
x =< 1/3
We have got the result x =< 1/3 for our consideration of x < 0
Anything equal to 1/3 is not negative and anything to the left of 1/3 need not be negative.
And as something negative is always less than 1/3, the verdict is x < 0
Answer: Either x >=1/3 or x<0
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