SOLUTION: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. What can you conclude about AC in ABC? AB + AC = 5; AC + BC = 4

(A)  We are given 

         AB + AC = 5,    (1)

         AC + BC = 4.    (2)


     Subtract equation (2) from equation (1).  You will get

         AB - BC = 5 - 4

     and then

         AB - BC = 1.    (3)


(B)  We have this triangle inequality (I use here one of the three similar inequalities)

         AB < AC + BC.


     It implies, after subtracting the length BC from both sides

         AB - BC < AC,

     or the same in this form

         AC > AB - BC.    (4)



(C)  But, according (3),  AB - BC = 1.  So, replace AB - BC in the right side of (4) by 1.

     You will get, as a consequence

         AC > 1.


ANSWER 1.  From the given information, we can conclude that  AC > 1.



               We can even make the answer stronger.  


For it, notice that from equation (2)  AC < 4.


So, we can write


ANSWER 2.  From the given information, we can conclude that  1 < AC < 4.