Problem Solving

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

Answer: B
Difficulty level: 600 - 650
Source: www.gmatprepnow.com

Brent@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

Answer: B

Cheers,
Brent