Inequality on open interval

In summary, the conversation discusses the inequality x > y-1 for all y \in ]0,1[ and whether it is possible to state that x > 0 holds. One person suggests that x \ge 0 holds, as y-1 can be in the range (-1,0) and x can be in the range [0,\infty). However, the strict inequality x > 0 is uncertain.
  • #1
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Given the fact that the following inequality must hold;

x > y-1 For all y[tex] \in[/tex] ]0,1[ (an open interval)

and given the fact that one can choose y After one chooses x, can one then state that x > 0 holds?

My idea was to say that at least x >= 0 holds because:

1) Someone picks a negative x that is arbitrarily close to 0, say -0.000...001.
2) I can now choose a y from the interval ]0,1[, say 0.999999... so that y-1 > x
3) Therefore nobody can pick a negative x so that the inequality holds

However, I am even more unsure about the strict inequality x > 0. It seems unlikely to me that it holds.

How do you properly reason about these kind of things?
 
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  • #2
If [itex]y \in (0,1)[/itex], then [itex]y-1 \in (-1,0)[/itex], so [itex]x \in [0,\infty)[/itex], that is, [itex]x \ge 0[/itex], satisfies the inequality. Note that this is a weaker condition than x>0, so both conditions hold.
 

What is an open interval?

An open interval is a set of real numbers that includes all numbers between two given values, but does not include the given values themselves. For example, the open interval (1, 5) includes all numbers between 1 and 5, but not 1 or 5.

What is inequality on an open interval?

Inequality on an open interval refers to the relationship between two values within an open interval. It means that one value is greater than or less than the other value, but not equal to it. For example, in the open interval (1, 5), the inequality 2 < 4 would be true, but the inequality 3 > 3 would be false.

Why is inequality on an open interval important?

Inequality on an open interval is important because it allows us to compare and analyze different values within a certain range. It can help us understand the distribution of data and identify patterns or trends. It is also used in many mathematical and scientific applications.

What is the symbol for inequality on an open interval?

The symbol for inequality on an open interval is < (less than) or > (greater than). These symbols are used to indicate which value is smaller or larger, respectively, within the open interval.

How can we solve inequality on an open interval?

To solve inequality on an open interval, we can use algebraic methods such as addition, subtraction, multiplication, and division to manipulate the given values and determine the relationship between them. Graphing or creating a number line can also help visualize the solution.

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