What reasoning error am I making here?

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In summary, the conversation discusses finding an upper bound for the expression (x + 2)/x on the interval (1, 4). The first method suggests using 4 and 1 as the top and bottom x values, respectively, while the second method incorporates the -5 term into the fraction. The correct upper bound is 8, but the book uses the second method to obtain an upper bound of 18. Both methods are valid, but the second one magnifies the result without a clear reason.
  • #1
blunkblot
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I apologize for a misclick double post:

Mentor action: double posted thread merged. No harm, no foul.
 
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  • #2
Apparently I cannot read. I figure this is the wrong thread. Please move or kill as needed :(

You need to find M where

[tex]\left|\frac{x+2}{x}-5\right| \le M, x \in (1, 4)[/tex]

To do this, I put 4 for the top x, 1 for the bottom x, to give the greatest quotient possible, plus |-5|, which finally gives 11. The correct answer is to combine 5 into the fraction with x's, and that gives a different answer.

What did I overlook by NOT combining them?

You can view the entry on google books preview here:

http://books.google.com/books?id=10... by first writing the given function"&f=false

Gratitude for enlightenment.
 
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  • #3
Look at the range of values for (x + 2)/x for x in [1, 4]. This function is defined at all points in this interval, and is decreasing on this interval. The largest value of (x + 2)/x on this interval comes when x = 1.
 
  • #4
Mark44 said:
Look at the range of values for (x + 2)/x for x in [1, 4]. This function is defined at all points in this interval, and is decreasing on this interval. The largest value of (x + 2)/x on this interval comes when x = 1.

This I understand. If you substitute x with 1, you can get | 3 | + | 5 | = 8 as an upper bound.

The solution as linked actually has M = 18. What makes 18 a better upper bound than 8? Or, is the book wrong?

Thank you
 
  • #5
It shouldn't be a better upper bound. I don't know how the book got 18, but it is a M that works, so it's technically correct as well.
 
  • #6
The book got it by merging the -5 into the fraction, giving

[tex]\left| \frac{-4x + 2}{x} \right| [/tex]

Then it maximizes this fraction within the allowed values for x by using 4 for the top x, 1 for the bottom x, and applying absolute value to each term since |-4x + 2| <= |4x| + |2|

This gives you 18. What makes this a good method?

It's really confusing, because by turning 5 into 5x/x, and plugging DIFFERENT values into the top x and bottom x, it greatly magnifies the result, apparently without good reason.
 
  • #7
Magnifying the result is a good enough reason, since you're looking for any upper bound. But your way is another perfectly fine way to find an upper bound.
 
  • #8
Makes sense, thank you.
 

Related to What reasoning error am I making here?

1. What is a reasoning error?

A reasoning error is a mistake or flaw in the logical process of reaching a conclusion or making a decision. It occurs when the reasoning used is incorrect or flawed, leading to an incorrect or irrational conclusion.

2. How can I identify a reasoning error?

You can identify a reasoning error by first understanding the basic principles of logic and reasoning. This includes understanding common logical fallacies and errors in thinking, such as circular reasoning, hasty generalizations, and false analogies. You can also examine your thought process and check for any assumptions or biases that may be influencing your reasoning.

3. What are some examples of common reasoning errors?

Some common examples of reasoning errors include confirmation bias, which is when you only seek out information that supports your existing beliefs, and post hoc fallacy, which is when you assume a cause-and-effect relationship between two events based on timing alone. Other examples include appeal to authority, slippery slope fallacy, and ad hominem attacks.

4. How can I avoid making reasoning errors?

To avoid making reasoning errors, it is important to think critically and objectively. This means considering all available evidence and information, questioning your assumptions and biases, and being open to changing your mind if new evidence or reasoning is presented. It can also be helpful to seek out alternative perspectives and actively listen to others' opinions.

5. Can reasoning errors be corrected?

Yes, reasoning errors can be corrected. The first step is recognizing and acknowledging that an error has been made. Then, one can re-examine the thought process and identify where the error occurred. Finally, one can use critical thinking skills to correct the error and reach a more logical and rational conclusion.

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