GeometryWhat is the range of possible values for x? The diagram is not to scale.

By the Hinge Theorem, we know that 
Angle ABD > Angle DBC

The "hinge" can be thought of the angle. A door opens and closes, which is attached to the hinge. The more the door opens, the larger the angle is. The value 11 is larger than 8, so the "door" so to speak is wider in angle. The larger the opposite side from an angle, the larger the angle itself. 

So when comparing the two angles 58 and (5x+3), the 58 degree angle is larger.

58 > 5x+3
58-3 > 5x
55/5 > x
11 > x
x < 11

At the same time, the angle DBC can't be negative or zero, so
5x + 3 > 0
5x > -3
x > -3/5
-3/5 < x

Together -3/5 < x and x < 11 combine to the answer of -3/5 < x < 11

This means x can be any number between -3/5 and 11; however, x cannot be equal to -3/5, nor can it be equal to 11