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If x is a positive integer, which of the following COULD : Problem Solving (PS)
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# If x is a positive integer, which of the following COULD

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If x is a positive integer, which of the following COULD [#permalink]
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GMATPrepNow wrote:
If x is a positive integer, which of the following COULD represent the lengths of the 3 sides of a triangle?

i) x, 2x + 2, x + 2
ii) 2x, 3x, 2x - 7
i) x/2, x/6, x/4

A) i only
B) ii only
C) iii only
D) i and ii only
E) ii and iii only

*kudos for all correct solutions

As it is given that x is a +ve int. we will put value of x as any positive integer and then we can look for the set which satisfies the condition of
sum of two sides> third side> difference between two sides of triangle
a+b>c>a-b
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Re: If x is a positive integer, which of the following COULD [#permalink]
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GMATPrepNow wrote:
If x is a positive integer, which of the following COULD represent the lengths of the 3 sides of a triangle?

i) x, 2x + 2, x + 2
ii) 2x, 3x, 2x - 7
i) x/2, x/6, x/4

A) i only
B) ii only
C) iii only
D) i and ii only
E) ii and iii only

*kudos for all correct solutions

Solution :-
a) Sum of two side is greater than third side.

1) a =x
b = 2x+2
c = x +2

a+b = x+2x+2 = 3x+2.
a+c = 2x+2 =b. not satisfied
b+c = 3x+4.

Could not be true.

2) a=2x
b=3x
c = 2x - 7.

Let us consider x = 4.
a+b>c ==> 5x>2x-7
a+c>b ==> 4x-7 > 3x
b+c>a ==> 5x-7 > 2x.

This is hard to break, but x=8 will satisfy all three inequalities.

3) a = x/2
b = x/6
c = x/4

Let us consider x = 12.
a = 12/2 = 6.
b = 12/6 = 2.
c = 12/4 = 3.

a+b >c ==> 8>3.
a+c > b ==> 9>2.
b+c > a ==> 5>6. Not satisfy.

Couldn't be.

So only 2nd satisfies this condition.

Ans - B
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Re: If x is a positive integer, which of the following COULD [#permalink]
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GMATPrepNow wrote:
If x is a positive integer, which of the following COULD represent the lengths of the 3 sides of a triangle?

i) x, 2x + 2, x + 2
ii) 2x, 3x, 2x - 7
i) x/2, x/6, x/4

A) i only
B) ii only
C) iii only
D) i and ii only
E) ii and iii only

There's a triangle property that says: (length of LONGEST side) < (SUM of the other two lengths)
So, for example, 2, 3, 6 CANNOT be the lengths of sides in a triangle, because 6 > 2 + 3

i) x, 2x + 2, x + 2
Since x is a positive integer, we can see that 2x+2 will be the LONGEST side.
Now compare this length to the SUM of the two other side lengths.
Is it true that 2x + 2 < x + (x + 2)?
Simplify: 2x + 2 < 2x + 2
This is NOT true.
So, x, 2x + 2, x + 2 CANNOT represent the lengths of the 3 sides of a triangle
Check the answer choices.... eliminate A and D

ii) 2x, 3x, 2x - 7
Since x is a positive integer, we can see that 3x will be the LONGEST side.
Now compare this length to the SUM of the two other side lengths.
Is it true that 3x < 2x + (2x - 7)?
Simplify: 3x < 4x - 7
This COULD be true.
For example, if x = 10, then we get: 3(10) < 4(10) - 7, which IS true.
So, 2x, 3x, 2x - 7 COULD represent the lengths of the 3 sides of a triangle
Check the answer choices.... eliminate C

iii) x/2, x/6, x/4
In this case, x/2 will be the LONGEST side.
Now compare this length to the SUM of the two other side lengths.
Is it true that x/2 < x/6 + 4/x?
Let's eliminate the fractions to make is easier for us to determine whether the inequality holds true.
Multiply both sides of the inequality by 12 to get: 6x < 2x + 3x
Simplify: 6x < 5x
Since x is POSITIVE, we can see that this inequality is NOT true.
So, x/2, x/6, x/4 CANNOT represent the lengths of the 3 sides of a triangle
Check the answer choices.... eliminate E

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If x is a positive integer, which of the following COULD [#permalink]
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GMATPrepNow wrote:
If x is a positive integer, which of the following COULD represent the lengths of the 3 sides of a triangle?
i) x, 2x + 2, x + 2
ii) 2x, 3x, 2x - 7
i) x/2, x/6, x/4
A) i only
B) ii only
C) iii only
D) i and ii only
E) ii and iii only

Option B. Its quite simple actually. Question asks which COULD be the answer. "Could" here means that for any value of x possible, which one of these satisfies the criterion of being the 3 sides of a triangle.
General condition for 3 line segments of length a, b, and c to be a triangle is that either a+b>c or |a-b|<c.

Lets take first case, if we add first and third lengths, we get x+ (x+2) = 2x+2. This is equal to the middle length. But, according to our general rule, addition of "ANY" 2 lengths should always be greater than the third one. Hence, it fails to qualify.
2nd case, adding first and third we get, 2x + (2x-7) => 4x-7. This however is tough to figure out whether 4x-7 is greater than 3x. Lets keep it as a contender.
3rd case, lets add up the 2 smaller ones. Obviously x/2 is biggest, so adding up x/6 and x/4 we get 5x/12. Obviously 5x/12 is smaller than x/2. Hence, it fails to qualify too.

Since we dont have "none of these" in the answer options, its easier to right away tick the answer b and move on. But, for the sake of getting 100% confident, lets take some values and see if it COULD be the answer.

If we take x = 2, 4x-7 gives 1 and 3x gives 6. Wrong!
But if we take a bigger value say 10, 4x-7 gives 33 and 3x gives 30. Hence sum of 2 is bigger than the third side and hence the answer.

PS: It would have been fun if there were a "none of these" option among the given options
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Re: If x is a positive integer, which of the following COULD [#permalink]
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bunny12345 wrote:
PS: It would have been fun if there were a "none of these" option among the given options

Good point! That would have made a good distractor.

Cheers,
Brent
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Re: If x is a positive integer, which of the following COULD [#permalink]
GMATPrepNow wrote:
bunny12345 wrote:
PS: It would have been fun if there were a "none of these" option among the given options

Good point! That would have made a good distractor.

Cheers,
Brent

Hehe yeah!

PS: I was also hoping for a kudo from you
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Re: If x is a positive integer, which of the following COULD [#permalink]
Hi BrentGMATPrepNow, had there been an option of None of these, the answer would have been None of these, right?
Please clarify, because, from condition 2, we can also infer x>2.3. Taking x=5 for instance does not make option 2 a triangle.
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Re: If x is a positive integer, which of the following COULD [#permalink]
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beeblebrox wrote:
Hi BrentGMATPrepNow, had there been an option of None of these, the answer would have been None of these, right?
Please clarify, because, from condition 2, we can also infer x>2.3. Taking x=5 for instance does not make option 2 a triangle.

The correct answer would still be B, since the question asks "which of the following COULD represent the lengths of the 3 sides of a triangle"
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Re: If x is a positive integer, which of the following COULD [#permalink]
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