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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]  26 May 2018, 09:51
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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
[Reveal] Spoiler: OA

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Re: If x is a positive integer, which of the following COULD [#permalink]  26 May 2018, 21:09
There are some triangle rules that apply for all triangles. One of that is the length of a side of a triangle is always less than the sum of other two sides of the triangle.

Let us test option i

x, 2x +2, x+2

If we add the first and the third side
x + x +2 = 2x+2 which is equal to the second side so this is false

Let us test option ii
2x, 3x, 2x - 7

Since the third side of this triangle includes a constant we can get some scenarios where the triangle can satisfy triangle properties and in some instance it does not satisfy the triangle property. However, this is a could be question so a single possibility is good enough to be true.

For eg. If the said integer is 1 we would have the three sides as,
2,3, -5
since length cannot be -ve this is not a possibility
However let us take x = 10 then,
20, 30, 13
This satisfys the triangle property

Let us test option iii
$$\frac{x}{2}, \frac{x}{6}, \frac{x}{4}$$
The two smallest sides are x/6 and x/2. Option iii can only be a triangle if x/2 < x/6 + x/4
$$\frac{x}{6} + \frac{x}{4} = \frac{2x + 3x}{12} = \frac{5x}{12}$$
Since $$\frac{x}{2}$$ > $$\frac{5x}{12}$$ option iii also is not correct.

option B
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Re: If x is a positive integer, which of the following COULD [#permalink]  28 May 2018, 06:42
Expert's post
GreenlightTestPrep 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 + x/4?
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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Re: If x is a positive integer, which of the following COULD   [#permalink] 28 May 2018, 06:42
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# If x is a positive integer, which of the following COULD

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