# Proofs regarding inequalities and number line stuff

• mscbuck
In summary, proving inequalities and understanding the number line allows us to accurately compare and represent numerical values, which is crucial in fields such as mathematics and science. This knowledge also helps in problem solving by allowing us to logically approach and solve complex mathematical problems. Common symbols used in proving inequalities are <, >, ≤, and ≥, and these can be represented by open and closed circles on a number line. To determine if an inequality statement is true or false, we must consider the relationship between the numbers. Understanding proofs regarding inequalities and the number line can also have practical applications in real-life situations, such as budgeting, comparing prices, and analyzing data.
mscbuck

## Homework Statement

"There is a very useful way of describing the points of the closed interval [a,b] (where we assume, as usual, that a < b)

a. Consider the interval [0,b]. Prove that if x is in [0,b] then x = tb for some t with 0 <= t <= 1. What is the significance of the number t? What is the mid-point of the interval [0,b]

b. Now prove that if x is in [a,b], then x = (1-t)a + tb for some t with 0 <= t <= 1. Hint: Can also be written as a + t(b-a). What is the midpoint of the interval [a,b]? What is the point 1/3 of the way from a to b?

c. Prove, conversely, that if 0 <= t <= 1, then (1-t)a + tb is in [a,b]

N/A

## The Attempt at a Solution

I think I'm really close and I'm just missing something extremely stupid. I solved part a like so
0 <= x <= b
0/b <= x/b <= b/b
0 <= x/b <= 1
Let x/b = t
0 <= t <= 1

I'm stuck on part b however. No matter how many times I did the algebra I just never could come up with the right equation. I assume to start out like this:
a <= x <= b
a/b <= x/b <= b/b
Let x/b = t
a/b <= t <= 1
a <= tb <= b
From here I'm somewhat stuck. I see that "tb" is part of the equation I"m looking for, I just need a (1-t)a from there.

I also think I have solved c:
0 <= t <= 1
0 * (b-a) <= t * (b-a) <= 1 (b-a)
0 <= t(b-a) <= b -a
a <= a + t(b-a) <= b

Last edited:
For (b), given interval [a, b], consider instead the interval [0, b-a]. You have, in part (a) shown that any point, x, in that interval, can be written in the form x= t(b- a) for some t between 0 and 1. Now "shift" that to put x between a and b.
(You are looking at b/a. Look at b- a instead.)

Thanks HallsofIvy!

Now I am on part d.) which states: The points of the OPEN interval (a,b) are those of the form (1-t)a + tb for 0 < t < 1.

Is there anything that is different about approaching this problem because of an open interval?

No difference at all. Except, of course, since t= 0 gives a and t= 1 gives b, you have to drop those values.

## 1. What is the purpose of proving inequalities and understanding the number line?

The purpose of proving inequalities and understanding the number line is to accurately represent and compare numerical values. This is especially important in mathematics and science, where precise measurements and relationships between numbers are crucial.

## 2. How do proofs regarding inequalities and the number line help in problem solving?

By understanding proofs regarding inequalities and the number line, we can apply this knowledge to solve complex mathematical problems. This allows us to logically and systematically approach problems and reach accurate solutions.

## 3. What are the common symbols used in proving inequalities and representing them on a number line?

The common symbols used in proving inequalities are <, >, ≤, and ≥. These symbols represent "less than", "greater than", "less than or equal to", and "greater than or equal to", respectively. On a number line, these symbols are represented by open and closed circles, with the closed circle representing the number being included in the inequality.

## 4. How do you know if an inequality statement is true or false?

An inequality statement is true if the relationship between the numbers is accurate. For example, if the statement is "3 > 2", this is true because 3 is greater than 2. On the other hand, if the statement is "5 < 4", this is false because 5 is not less than 4.

## 5. How can understanding proofs regarding inequalities and the number line help in real-life situations?

Understanding proofs regarding inequalities and the number line can help in real-life situations such as budgeting, comparing prices, and analyzing data. By being able to accurately compare and represent numerical values, we can make informed decisions and solve real-world problems.

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