Find all numbers x which satisfy the following inequality

[r0bHadz]

Homework Statement

(1/x) + (1/(1-x)) > 0

Homework Equations

The Attempt at a Solution

1+x-x/(x-x^2) > 0
1/(x-x^2) > 0
x-x^2 > 0
x> x^2 only occurs when 0<x<1

but in the solutions Spivak tells me

"x>1 or 0<x<1"

 

[BvU]

And if you try it for some $x>1$ you see that Spivak may well be right. Where do you suspect the flaw in your reasoning ?

Oops: [Edit] needed: if you try it for some $x>1$ you see that Spivak may well be wrong

(Provided you gave a truthful rendering of the problem statement and the corresponding solution !)

By the way: some brackets needed here ! :

1+x-x/(x-x^2) > 0

 

[Ray Vickson]

Homework Statement

(1/x) + (1/(1-x)) > 0

Homework Equations

The Attempt at a Solution

1+x-x/(x-x^2) > 0
1/(x-x^2) > 0
x-x^2 > 0
x> x^2 only occurs when 0<x<1
but in the solutions Spivak tells me

"x>1 or 0<x<1"

When you write 1+x-x/(x-x^2) > 0 you are saying
$$1+x-\frac{x}{x-x^2} > 0,$$
which is wrong. If you mean to say that
$$\frac{1+x-x}{x-x^2} > 0$$ then you need to use parentheses, like this: (1+x-x)/(x-x^2) > 0.

 

[epenguin]

The problem statement represents a quite easy problem. I can't see what connection there is between the problem statement in the first line and the calculation you do. Start again- it is just straightforward combination of fractions into one and then the answer is almost obvious.

 

[r0bHadz]

The problem statement represents a quite easy problem. I can't see what connection there is between the problem statement in the first line and the calculation you do. Start again- it is just straightforward combination of fractions into one and then the answer is almost obvious.

I don't understand how he got x>1 from the problem statement

 

[Ray Vickson]

I don't understand how he got x>1 from the problem statement

He got it by making a mistake; your answer is the correct one.

 

[FactChecker]

The problem statement represents a quite easy problem. I can't see what connection there is between the problem statement in the first line and the calculation you do.

He omitted some necessary parenthesis and should have shown some intermediate steps.

@r0bHadz , you should get in a habit of doing calculations in small, very safe, steps and using parenthesis whenever there is ambiguity. Do not try to do too much in your head. The calculations will go much easier and more reliably.:
1/x + 1/(1-x) = (1-x)/(x(1-x)) + x/(x(1-x)) = (1-x+x)/(x(1-x)) = 1/(x(1-x))

PS. It is very common for me to see beginners trying to do more in their head than I do. And I have a PhD in math.

 

[SammyS]

...

but in the solutions Spivak tells me

"x>1 or 0<x<1"

That is the solution for something like $\ \displaystyle \frac 1 x + \frac 1 {|1-x|} > 0 \,.$

 

[epenguin]

Ah I see, the calculation is simple but Spivak's x>1 is a mistake.

 

[FactChecker]

Ah I see, the calculation is simple but Spivak's x>1 is a mistake.

Or, as @SammyS points out, some absolute value signs may have been misprinted / misinterpreted as parentheses.