the problem shows a larger triangle that has been divided into two smaller ones, via a spliting of an acute angle into two different size angles one 52 degrees and the other one 31 degrees.
Notice as well that these two smaller triangles have two congruent sides :
AC = AB, and alsoa common side AD.
Then, we can say that the angle wich measures 52 degrees and opposes side BD (which measures 12) has a bigger opening than the angle 31 degrees, and bearing as well the same length of the sides that conform those angles, is definitely going to have a LARGER opposite side than angle 31 degrees.
Then we can write the following first inequality:
opposite side to 52 degrees > opposite side to 31 degrees, which in math expression using the sides measures, give:
12 > 2 x - 6
and we proceed to solve for x in the inequality
add 6 to both sides
12 + 6 > 2 x
divide by 2 both sides (notice that since 2 is positive, the division doesn't change the direction of the inequality)
18 / 2 > x
9 > x
which is the same as:
x < 9
So we have one of the endpoints.
For the other limit for the smaller side (to the left) we use the fact that for sure, the side opposed to 31 degrees angle must be larger than "0" (zero) independent of the lengths of sides AC and AD. So we write this and use the expression for side DC that they give us:
2 x - 6 > 0
add 6 to both sides
2 x > 6
divide both sides by positive 2
x > 6 / 2
x > 3
So, now we put together the conditions for the values we found for the minimum and maximum that x can reach:
3 < x < 9
Please, type these answers in the provided boxes.