Absolute problem!

x = 2a-1 => |x|+|x+8| = 8 => x between -8 and 0 => x in {-7,-5,-3,-1}

The sure way of solving this problem is plotting out the graphs, as I have done here. The range $-3.5 \le a \le 0.5$ satisfies the equation. Therefore, the number of integers that satisfy the equation is $\boxed{4}$ and they are $-3,-2,-1$ and $0$ .

The solution in integers is a = 0, -1, -2, and -3. It is worth noting that there are also 5 rational solutions also: 1/2, -1/2, -3/2, -5/2, and -7/2. Ed Gray