Question 1047889
.
in a triangle ABC, AB=x+2, BC=2x+3,AC=3x-5. Find all possible values of x
~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
To solve the problem you need to solve all three "triangle inequalities":

AB + BC > AC,   i.e.  (x+2)  + (2x+3) > 3x-5,    (1)
AB + AC > BC,   i.e.  (x+2)  + (3x-5) > 2x+3,    (2)   and
BC + AC > AB,   i.e.  (2x+3) + (3x-5) > x+2.     (3)


(1) is equivalent to 5 > -5, which is always true and doesn't carry useful information;

(2) is equivalent to 4x-3 > 2x+3, i.e. 2x > 6, which means x > 3.

(3) is equivalent to 5x-2 > x+2, i.e. 4x > 4, which means x > 1.


So, you solved the problem. All possible values of x are x > 3.
It is the necessary and sufficient condition.


<U>Answer</U>. All possible values of "x" are determined by only one condition  x > 3.
</pre>