If the sides of a triangle have measurements 3x + 4 , 6x - 1, and 8x + 2, find all possible values of x.
I also need help with this question please
Thanks anyway but that isn't one of the choices the answer is x> 3/11 i figured it out thanks for trying :)
All sides must be greater of zero.
3x+4 always>0
8x+2 always>0
Only side: 6x-1 can be negative.
That's why:
6x-1>0
6x>1 Divide both sides with 6
x>1/6
Anonymus solution not completely.
All sides must be greater of zero:
3x+4>0
3x> -4 Divide with 3
x> -4/3
6x-1>0
6x>1 Divide with 6
x>1/6
8x+2>0
8x> -2 Divide with 8
x> -2/8
x> -1/4
Least of that numbers is -4/3= -1.3333
x> -4/3 is solution
x>3/11 is also > -4/3
All x> -4/3 is choices
The condition that all sides have to be positive, as Anonymous used, is not sufficient.
In any triangle the sum of 2 sides must be greater than the third side, so
3x+4 + 6x-1 > 8x + 2 ----> x > -1
AND
3x+4 + 8x + 2 > 6x-1 ---> x > -7/5
AND
8x + 2 + 6x - 1 > 3x+4 --> x > 3/11
the intersection of all three conditions is
x > 3/11
To show that Anonymous is incorrect, pick a value of x between his/her answer of 1/6 and mine of 3/11
e.g. x = 11/50
3x+4 --> 4.66
6x-1 --> .32
8x+2 --> -.24 , contradiction
I have a value of x > 1/6 which did not work.
To find the possible values of x, we need to apply the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, let's apply the inequality to the given measurements of the triangle:
3x + 4 + 6x - 1 > 8x + 2 (1)
3x + 4 + 8x + 2 > 6x - 1 (2)
6x - 1 + 8x + 2 > 3x + 4 (3)
Simplifying each inequality, we get:
9x + 3 > 8x + 2 (4)
11x + 6 > 6x - 1 (5)
14x + 1 > 3x + 4 (6)
Let's solve each inequality to find the possible values of x:
(4) Subtract 8x from both sides and subtract 3 from both sides:
9x - 8x > 2 - 3
x > -1
(5) Subtract 6x from both sides and subtract 6 from both sides:
11x - 6x > -1 - 6
5x > -7
Divide both sides by 5, remembering to reverse the inequality when dividing by a negative number:
x < -7/5
(6) Subtract 3x from both sides and subtract 1 from both sides:
14x - 3x > 4 - 1
11x > 3
Divide both sides by 11:
x > 3/11
Putting the values together, we have:
-1 < x < 3/11
Therefore, the possible values of x are all real numbers between -1 and 3/11 (exclusive).