How many solutions does a triangle with side lengths a = 4, A = 68, and b = 10 have?

The easiest way to find if a triangle can be formed is by applying the law of sinus: a/sin (A) = b/sin (B). The range of sin must be [-1, 1].

In our problem:

sin68/4 = sin (B) / 10

sin (B) = 10/4*sin68 = 2.5sin (68)

Function sin is an increasing function between 0-90 degrees so we can say that 30<68<90 so do sin (30)
1/2 < sin (68) < 1. Multiplying by 2.5 we have:

1.25<2.5sin (68) < 2.5

1.25
But as I said the range of sin must be [-1, 1]. So there is no solution in order to form a triangle (No triangle can be formed).