If x is a positive integer, and two sides of a certain triangle have lengths 5x+1 and x respectively, which of the following could be the length of the third side of the triangle?
I. x + 1
II. x + 0
III. 8x  Choose the correct choice from the following answers.
A) I only B) II only C) I and II only D) II and III only E) I, II and III
Let the third side of triangle be AB   Triangle rule states that:   Difference of other 2 sides &lt; any side &lt; sum of other two sides   So 5x+1 - (x) &lt; AB &lt; 5x+1 + (x)   i.e. 4x + 1 &lt; AB &lt; 6x + 1   Let's take x = 1 :   So 5 &lt; AB &lt; 7   Check if the options lies between 5 and 7.   I: x+1 = 2 ; could be a possible value.   II: x+0 = 1 ; could be a possible value.   III: 8x = 8; could not be a possible value.   I and II could be a possible value.   Answer: I and II only
<result>C</result>
If x is a positive integer, and two sides of a certain triangle have lengths 5x+1 and x respectively, which of the following could be the length of the third side of the triangle?
I. x + 1
II. x + 0
III. 8x