Henry, whose mass is 95 kg, stands on a bathroom scale in an elevator. The scale reads 830 N for the first 3.8 s after the elevator starts moving, then 930 N for the next 3.8 s. What is the elevator's speed 7.6 s after starting?

When standing on the bathroom scale within the moving elevator, there are two forces acting on Henry's mass: Normal force and gravity.
Gravity is always downward, and normal force is perpendicular to the surface on which the mass is located (the bathroom scale), in upward direction.
Normal force, can adopt any value needed to match the acceleration of the mass, according to Newton's 2nd Law.
Gravity (which we call weight near the Earth's surface) can be calculated as follows:

[tex]F_{g} = m*g = 95 kg * 9.8 m/s2 = 930 N (1)[/tex]

[tex]F_{net} = F_{g} -F_{n} = m*a\\ F_{net} = 930 N - 830 N = 100 N = 95 Kg * a[/tex]

As the gravity is larger than normal force, this means that the acceleration is downward, so, we choose this direction as the positive.

[tex]a =\frac{F_{net} }{m} =\frac{100 N}{95 kg} = 1.05 m/s2[/tex]

We can find the speed after the first 3.8 s (assuming a is constant), applying the definition of acceleration as the rate of change of velocity:

[tex]v_{f} = a* t = 1.05 m/s * 3.8 m/s = 4.0 m/s[/tex]

Now, if during the next 3.8 s, normal force is 930 N (same as the weight), this means that both forces are equal each other, so net force is 0.
According to Newton's 2nd Law, if net force is 0, the object is either or at rest, or moving at a constant speed.
As the elevator was moving, the only choice is that it is moving at a constant speed, the same that it had when the scale was read for the first time, i.e., 4 m/s downward.