For what value of x does (1/x) + (1/(1-x)) > 0

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In summary, the conversation discusses finding the value of x for which the equation 1/x + 1/(1-x) is greater than 0. It is determined that x cannot equal 0 or 1 and that the equation is positive when 0 < x < 1. The solution manual states the answer as 0 < x < 1 or x > 1, but it is agreed that this is incorrect and the correct answer is 0 < x < 1.
  • #1
zeion
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Homework Statement



For what value of x does [tex] \frac{1}{x} + \frac{1}{1-x} > 0 [/tex]

Homework Equations


The Attempt at a Solution



So I see that x not = 0 and x not = 1.
Then I added the fractions and see that it will only be positive if the bottom is positive
ie. x(1-x) > 0
I define f(x) = x(1-x) that is a parabola.
It will change signs at x = 0 and x = 1.
So I test some values in the intervals and see that it is positive when 0 < x < 1
So that is the answer.
But the solution says that it is 0 < x < 1 or x > 1?
If I put in 2 > 1 I will get (1/2) + (1/(1-2)) = (1/2)+(1/(-1) = -1/2 not > 0
 
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  • #2
Solution manuals do make mistakes. It sounds like you have a good handle on this proof technique, and have taken advantage of some sanity checks (another would be to graph 1/x + 1/(1-x)) that support your answer and conflict with the manuals, so you would have good reason to remain confident in your solution.
 
  • #3
There's nothing to worry about,
Your answer is correct, there must have been some printing error in the solution.
 
  • #4
It seems that the solution manual is incorrect.

edit: began my original response before others posted. Carry on...
 
  • #5
Zeion pwned the textbook.
 

Related to For what value of x does (1/x) + (1/(1-x)) > 0

What is the significance of finding the value of x in the inequality (1/x) + (1/(1-x)) > 0?

The value of x is important because it represents the critical point at which the inequality changes from being less than zero to being greater than zero. This value can provide insights into the behavior of the expression and can help in solving related problems.

Why is it necessary to exclude certain values of x when solving the inequality (1/x) + (1/(1-x)) > 0?

It is necessary to exclude certain values of x because they can lead to a division by zero, which is undefined. In this inequality, the values of x=0 and x=1 must be excluded to avoid this issue.

What is the relationship between the values of x and the graph of the expression (1/x) + (1/(1-x))?

The values of x determine the behavior of the graph of the expression. For values of x less than 0 or greater than 1, the graph is below the x-axis and the inequality is not satisfied. For values of x between 0 and 1, the graph is above the x-axis and the inequality is satisfied.

Can the inequality (1/x) + (1/(1-x)) > 0 be solved algebraically?

Yes, the inequality can be solved algebraically by finding the critical points and testing intervals to determine when the inequality is satisfied. This method can provide an exact solution for the value of x that satisfies the inequality.

What is the practical application of solving the inequality (1/x) + (1/(1-x)) > 0?

The practical application of solving this inequality lies in understanding the behavior of certain mathematical expressions and functions. It can also be used in real-world scenarios, such as determining the optimal value for a variable in a mathematical model or analyzing the behavior of a physical system.

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