For the quadrilateral shown, how many different whole numbers could be the length of the diagonal represented by the dashed line?

Using the triangle inequality which says that the sum of any two sides of a triangle is greater than the remaining side, we have two triangles to consider

Top triangle....

22 + 18   = 40

So the greatest integer value that we can have for the dashed line is 39

18 + x  must be greater than 22

So the least integer value that we can have for the dashed line is 5

So the interval  that  solves  the dashed line in the top riangle is  [ 5, 39]     (1)

Bottom triangle

21 + 25 =  46

So the greatest integer value that we can have for the dashed line is  45

21 + x  = must be greater than 25

So the least integer value that we can have for the dashed line is 5

So the interval that solves the dashed line in the bottom triangle is  [5, 45 ]    (2)

Taking the intersection of (1)  and (2)  we have  that the   interval that solves the problem is

[ 5, 39 ]

So....the number of whole numbers that satisfies  the dashed line length  is

39  - 5  +  1   =

34  +  1  =

35