Question 828701
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Hi, there--

THE PROBLEM:
The sides of a triangle all have integral measure and can be represented by the expressions x 
+ 10, 2x -3, and 4x. For how many values of x will the sum of the two longer sides be 
divisible by the shortest side?

A SOLUTION:
The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

We have 3 situations that must all be true.
I. x + 10 < (2x - 3) + (4x)
II. 2x-3 < (x + 10) + (4x)
III. 4x < (x + 10) + (2x - 3)

CASE I:
x + 10 < 2x - 3 + 4x
x + 10 < 6x - 3
x < 6x - 13
-5x < -13
x > 13/5 (Remember to reverse the direction of the inequality.)


CASE II:
2x - 3 < x + 10 + 4x
2x - 3 < 5x + 10
-3x - 3 < 10
-3x < 10 + 3
-3x < 13
x > 13/-3 (Reverse inequality.)
x > -13/3

CASE III: 
4x < 2x - 3 + 10 + 4x
4x < 3x + 7
x < 7

Now we combine the information from each inequality Recall that the sides of this triangle 
have integral measure. Also, because it is the side length of the triangle.

x > 5/13 AND x > -13/3 ANG x < 7

x must be an integer such that x>13/5, so x = 3, 4, 5…
AND
x is an integer such that x>-13/3, so x = 1, 2, 3,…
AND
x is an integer such that x<7, so x = 1, 2, 3, 4, 5, 6, 7

Taken together, we see that  must equal 3, 4, 5, or 6 in order for the triangle to exist. 

For which values of x will the sum of the longer sides be divisible by the shortest side?

When x = 3,
x + 10 = 13
2x - 3 = 3
4x = 12
The sum of the longer sides is 25 which is not divisible by 3.

When x = 4,
x + 10 = 14
2x - 3 = 5
4x = 16
The sum of the longer sides is 30 which is divisible by 5.

When x = 5,
x + 10 = 15
2x - 3 = 7
4x = 20
The sum of the longer sides is 35 which is divisible by 7

When x = 6,
x + 10 = 16
2x - 3 = 9
4x = 24
The sum of the longer sides is 40 which is not divisible by 9.

For two values, x=4 and x=5, the sum of the longer sides is divisible by the shorter side.

Interesting problem. I hope this helps! Feel free to email if you have any questions about the 
solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com
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