Determine the set of values of ##n## that satisfy the inequality

In summary: SammyS let me look at this on Saturday... cheersIn summary, the conversation discusses finding a solution for a problem involving positive integer values. The approach is to determine the inequality ##\dfrac{n^2-1}{2}≥1## and the realization that ##0## is an integer but not a positive integer. It is then stated that ##n≥\sqrt 3## and it is suggested to use subtraction to show that ##\dfrac {n^2+1}{2}## is greater than ##\dfrac {n^2-1}{2}##. Finally, it is mentioned that the solution requires that ##n## must be an odd integer, but the approach to showing
  • #1
chwala
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Homework Statement
See attached
Relevant Equations
Inequalities
1653996550318.png


Find solution here;

1653996590449.png


Ok i just want clarity for part (a),
My approach is as follows, since we want positive integer values that satisfy the problem then,
##\dfrac {n^2-1}{2}≥1## I had earlier thought of ##\dfrac {n^2-1}{2}≥0## but realized that ##0## is an integer yes but its not a positive integer.
Therefore,
##\dfrac {n^2-1}{2}≥1, n^2-1≥2, n^2≥3 ⇒n≥\sqrt 3##

For,
##\dfrac {n^2+1}{2}≥, n^2≥1⇒n≥1##. The inequality satisfying the two is

##n≥\sqrt 3##

...Any insight on this is appreciated.

Part (b) is easy...no problem there...
 
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  • #2
The typed up image has a lot of "or"s that probably are supposed to be "and"s. Is that an official grading rubric? I would give it half points at most.

It's kind of lame to not just name the smallest integer that you computed, which is 2, not ##\sqrt{3}##. But just to make sure, ##(2^2-1)/2=1.5## is not an integer, so there is more work to be done.
 
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  • #3
Office_Shredder said:
The typed up image has a lot of "or"s that probably are supposed to be "and"s. Is that an official grading rubric? I would give it half points at most.

It's kind of lame to not just name the smallest integer that you computed, which is 2, not ##\sqrt{3}##. But just to make sure, ##(2^2-1)/2=1.5## is not an integer, so there is more work to be done.
Yes, its an official ms guide from a past paper...
 
  • #4
Office_Shredder said:
The typed up image has a lot of "or"s that probably are supposed to be "and"s. Is that an official grading rubric? I would give it half points at most.

It's kind of lame to not just name the smallest integer that you computed, which is 2, not ##\sqrt{3}##. But just to make sure, ##(2^2-1)/2=1.5## is not an integer, so there is more work to be done.
True, i had ##2## in my original working and thought i had done something wrong! Since ##n≥\sqrt 3⇒n≥2,nε\mathbb{z}^{+}##
 
  • #5
Great. What about the part where not all choices of ##n\geq 2## satisfy this? You have the positive part down, what about the integer part?
 
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  • #6
Office_Shredder said:
Great. What about the part where not all choices of ##n\get 2## satisfy this? You have the positive part down, what about the integer part?
I hope i am getting you right, we have ##\sqrt 3=±1.73205...## We will not consider the negative values of ##n## as the question only requires us to find positive integer values of ##n## . In that case the value of ##n=-1## will not be a solution.
 
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  • #7
chwala said:
Homework Statement:: See attached
Relevant Equations:: Inequalities

View attachment 302150

Find solution here;

View attachment 302151

Ok i just want clarity for part (a),
My approach is as follows, since we want positive integer values that satisfy the problem then,
##\dfrac {n^2-1}{2}≥1## I had earlier thought of ##\dfrac {n^2-1}{2}≥0## but realized that ##0## is an integer yes but its not a positive integer.
Therefore,
##\dfrac {n^2-1}{2}≥1, n^2-1≥2, n^2≥3 ⇒n≥\sqrt 3##
At this point, you have given the requirement for ##\dfrac {n^2-1}{2}## to be positive.

It should be very easy to show that ##\dfrac {n^2+1}{2} \gt \dfrac {n^2-1}{2}## for all ##n##, so the requirement on ##n## remains the same. I suggest using subtraction.

Seems like it's still left to show that to get integer results, we need that ##n## must be odd.
 
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  • #8
Yeah. I'm referring to the part where only odd n give actual integers. You haven't mentioned it at all in your posts.
 
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  • #9
Office_Shredder said:
Yeah. I'm referring to the part where only odd n give actual integers. You haven't mentioned it at all in your posts.
True, i did not mention that...i guess the ms is clear on that part...
 
  • #10
chwala said:
True, i did not mention that...i guess the ms is clear on that part...
Yes, it is true that the ms (marking scheme - or whatever) is clear in stating that ##n## must be an odd integer. What's not so clear in the ms, is how to show that.

What's even less clear is your approach to showing that ##n## must be odd.
 
  • #11
SammyS said:
Yes, it is true that the ms (marking scheme - or whatever) is clear in stating that ##n## must be an odd integer. What's not so clear in the ms, is how to show that.

What's even less clear is your approach to showing that ##n## must be odd.
Ok, I'll look at that later...cheers.
 
  • #12
chwala said:
Ok, I'll look at that later...cheers.
Maybe ... start with :

Suppose ##\displaystyle \dfrac{n^2-1}{2}=k, \text{ for }k\text{ a positive integer.}## Of course, you have already determined the condition for the positive part of the requirement.
 
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  • #13
SammyS said:
Maybe ... start with :

Suppose ##\displaystyle \dfrac{n^2-1}{2}=k, \text{ for }k\text{ a positive integer.}## Of course, you have already determined the condition for the positive part of the requirement.
@SammyS let me look at this on Saturday... cheers
 

Related to Determine the set of values of ##n## that satisfy the inequality

1. What is the meaning of "set of values" in this context?

The set of values refers to the range of possible numerical solutions that satisfy the given inequality. These values can be any real numbers that make the inequality true.

2. How do I determine the set of values that satisfy the inequality?

To determine the set of values, you will need to solve the inequality for n. This can be done by using algebraic methods such as isolating the variable or using properties of inequalities.

3. Are there any restrictions on the values of n?

It depends on the specific inequality given. Some inequalities may have restrictions, such as n being a positive integer or n being greater than a certain number. It is important to carefully analyze the given inequality to determine if there are any restrictions on n.

4. Can there be more than one set of values that satisfy the inequality?

Yes, there can be multiple sets of values that satisfy the inequality. This is because inequalities represent a range of values rather than a single value. You may need to use different methods or approaches to determine all possible sets of values.

5. How do I check if a given value of n is in the set of values that satisfy the inequality?

To check if a given value of n is in the set of values, simply substitute the value into the original inequality and see if it makes the inequality true. If it does, then the value is in the set of values that satisfy the inequality. If it does not, then the value is not in the set of values.

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