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1. In the triangle shown, the side lengths are given below in terms of a real-valued variable $x$. Find the range of all possible values of $x$, writing your answer in interval notation.

https://latex.artofproblemsolving.com/6/c/a/6ca2d738712c186c784acd9ae288810db5edaadd.png

 

2.One side of a triangle has length $10$, and the triangle has integer perimeter. What is the smallest possible perimeter of the triangle?

 

3.Two side lengths of a triangle are $11$ and $17$. What is the longest possible integer length of the third side of the triangle?

 Jan 28, 2020
 #1
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1. The set of possible x is [3,25].

 

2.  The smallest possible perimeter is 23.

 

3. The largest possible third side is 29.

 Jan 28, 2020
 #2
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Thanks!  I also need help with these problems:

 

Get lost guest  and put your effort into doing your own homework.

 Jan 28, 2020
edited by Melody  Feb 3, 2020
 #3
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I'm not certain what the "$" on each side of "x" represent.

 

If you are just considering all real numbers for the possibile replacements for x in the triangle whose sides are x + 15, 2x + 15, and 4x + 15, then, since the sum of any two sides of a triangle must be greater than the third side, there are these three possibilities:

 

I.    (x + 15) + (2x + 15) > 4x + 15     --->     3x + 30 > 4x + 15     --->     15 > x        --->     x < 15

II.   (x + 15) + (4x + 15) > 2x + 15     --->     5x + 30 > 2x + 15     --->     3x > -15     --->     x > -5

III.  (2x + 15) + (4x + 15) > x + 15     --->     6x + 30 > x + 15       --->     5x > -15     --->     x > -3    

 

The answer is (-3, 15)     [Using a value of -4 or -5 for x results in a side of the triangle being negative.]

 

Using "$x$ to refer to just "x", I can't get any possible of answers that work for questions 2 and 3.

 Jan 28, 2020
 #4
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I would appreciated if you put the latex into the latex translator.

 Jan 29, 2020

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