Is b less than or equal to 8 in this advanced equation?

  • Thread starter silenzer
  • Start date
In summary: Standard methods give the interval you questioned: -\sqrt{8}\le b\le\sqrt{8}\,.The "standard methods" are: pq< 0 if and only if p and q are of different sign: either "p< 0 and q> 0" or "p> 0 and q< 0". Since b^2- 8= (b-\sqrt{8})(b+\sqrt{8}), for that to be less than 0 we must have one of (a) b- \sqrt{8}> 0 and b+ \sqrt{8}< 0 or(b) b- \sqrt{8}
  • #1
silenzer
54
0

Homework Statement


This isn't going to be this complicated that I have to section my problem down to segments, this is just a part of an advanced problem. There is also something wrong with my browser, I can only type into a textbox where I start typing at the beginning. I can't press enter or anything like that.The problem is this: I cannot understand why this happens: http://j.mp/q2ySH9. I would think that b would be less or equal than BOTH of the numbers... yet it becomes an interval. EDIT: sorry, in the second step it's supposed to be b, not b squared.

Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution


Homework Statement


Homework Equations


The Attempt at a Solution

 
Last edited by a moderator:
Physics news on Phys.org
  • #2
silenzer said:

Homework Statement


This isn't going to be this complicated that I have to section my problem down to segments, this is just a part of an advanced problem. There is also something wrong with my browser, I can only type into a textbox where I start typing at the beginning. I can't press enter or anything like that.The problem is this: I cannot understand why this happens: http://j.mp/q2ySH9. I would think that b would be less or equal than BOTH of the numbers... yet it becomes an interval. EDIT: sorry, in the second step it's supposed to be b, not b squared.
[URL]http://j.mp/q2ySH9[/URL]
(Now your image is visible.)

[itex]b^2\le8[/itex][itex]b^2-(\sqrt{8})^2\le0[/itex][itex](b-\sqrt{8})(b+\sqrt{8})\le0[/itex]

Standard methods give the interval you questioned: [itex]-\sqrt{8}\le b\le\sqrt{8}\,.[/itex]
 
Last edited by a moderator:
  • #3
The "standard methods" are: pq< 0 if and only if p and q are of different sign: either "p< 0 and q> 0" or "p> 0 and q< 0".

Since [itex]b^2- 8= (b-\sqrt{8})(b+\sqrt{8})[/itex], for that to be less than 0 we must have one of
(a) [itex]b- \sqrt{8}> 0[/itex] and [itex]b+ \sqrt{8}< 0[/itex] or
(b) [itex]b- \sqrt{8}< 0[/itex] and [itex]b+ \sqrt{8}> 0[/itex]

(a) gives [itex]b> \sqrt{8}> 0[/itex] and [itex]b< -\sqrt{8}< 0[/itex] but those cannot both be true. Therefore, we must have (b) which gives [itex]b< \sqrt{8}[/itex] and [itex]b> -\sqrt{8}[/itex] or [itex]-\sqrt{8}< b< \sqrt{8}[/itex].

Since the initial problem had "[itex]\le 0[/itex]" rather than just "<", we must have
[tex]-\sqrt{8}\le b\le \sqrt{8}[/tex].

By the way, since 8= 4(2) and 4 is a "perfect square", you can write [itex]\sqrt{8}= 2\sqrt{2}[/itex] and that is typically preferred:
[tex]-2\sqrt{2}\le b\le 2\sqrt{2}[/tex]
might be preferred as an answer.
 

1. What does "B less or equal than 8" mean?

It means that the value of variable B is either less than or equal to 8. In other words, B can be any number from negative infinity up to and including 8.

2. How can "B less or equal than 8" be represented mathematically?

This statement can be represented as B ≤ 8, where the symbol ≤ means "less than or equal to" in mathematics.

3. Is "B less or equal than 8" an inclusive or exclusive statement?

It is an inclusive statement, meaning that B can be equal to 8 as well as any number less than 8.

4. Can a negative number satisfy the condition "B less or equal than 8"?

Yes, a negative number can satisfy this condition. For example, B could be -5, which is less than 8 and therefore meets the criteria.

5. How is "B less or equal than 8" different from "B less than 8"?

The main difference is that "B less or equal than 8" includes the possibility of B being equal to 8, while "B less than 8" does not. In other words, "B less or equal than 8" is a less strict condition than "B less than 8".

Similar threads

  • Precalculus Mathematics Homework Help
Replies
14
Views
358
  • Precalculus Mathematics Homework Help
Replies
10
Views
889
  • Precalculus Mathematics Homework Help
Replies
6
Views
2K
  • Precalculus Mathematics Homework Help
Replies
6
Views
801
  • Precalculus Mathematics Homework Help
Replies
18
Views
1K
  • Precalculus Mathematics Homework Help
Replies
7
Views
824
  • Precalculus Mathematics Homework Help
Replies
18
Views
2K
  • Precalculus Mathematics Homework Help
Replies
2
Views
620
  • Precalculus Mathematics Homework Help
Replies
8
Views
9K
  • Precalculus Mathematics Homework Help
Replies
4
Views
881
Back
Top