For all z, [z] denotes the least integer greater than or equal to z. Is [x]=0?
(1) -1 < x < -0.1
(2) [x + 0.5] = 1

ASIDE: What does "the least integer greater than or equal to z" mean?
Let's examine [2.4]
There are many integers that are greater than or equal to 2.4
They include 3, 4, 5, 6 . . .
The LEAST integer that's greater than or equal to 2.4 is 3.
So, [2.4] = 3

Likewise, [4.1] = 5
[11] = 11
and [-9.7] = -9

Now onto the question . . .

Target question: Is [x]=0?

Statement 1: -1 < x < -0.1
As you can see, if -1 < x < -0.1, then the integers that are greater than x include 0, 1, 2, 3, etc
The smallest of these integers is 0
So, it must be the case that [x] = 0
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: [x + 0.5] = 1
There are several values x that meet this condition. Here are two:
Case a: x = -0.1, in which case [x] = 0
Case b: x = 0.1, in which case [x] = 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer = A

Cheers,
Brent