Finding b (knowing that b>=0) in which Pr(2-b < Z < 2+b) = 0.4 and Z~Normal(0,1). Is it possible to find the value of b (without using a numerical method -  this observation weren't in the first version of the question) in which 
Pr(2-b < Z < 2+b) = 0.4
where b is a nonnegative value (b >= 0) and Z is a continuous random variable following a standard normal distribution? 
Would be nice find b in function of something like the inverse of the CDF of Z... 
Thanks in advance, 
Felipe
 A: This can easily be done numerically; Note that $P(2 - b < Z < 2 + b) = \Phi(2 + b) - \Phi(2 - b)$. Now any algorithm to solve equations approximately [e.g. Newton method] will yield a value of $b \approx  1.74688$ [see here].
However, I doubt that this equation can be solved exactly.
A: The answer to my original question is given below.
Pr(2-b < Z < 2+b) = 0.4 
Pr(-b < Z-2 < b) = 0.4
Pr((Z-2)^2 < b^2) = 0.4
(Z-2)^2 follows a non-central qui-squared random variable with 1 degree of freedom and non-centrality parameter given by 2^2 = 4. So let's call X = (Z-2)^2, then
Pr(X < b^2) = 0.4
F(b^2) = 0.4
Where F is the c.d.f. of a non-central qui-squared random variable with 1 degree of freedom and non-centrality 4, then
b^2 = invF(0.4)
Where invF(a) is the a-quantile of a non-central qui-squared random variable with 1 degree of freedom and non-centrality 4, then
b = squareroot[invF(0.4)]
invF(0.4) can be easily calculated using any stat software such as R [in R, just use the code qchisq(0.4,1,4)]. this gives invF(0.4) = 3.051606, then
b = squareroot[3.051606]
Taking the square root of 3.051606, finally we have 
b = 1.746885
Note that what is very interesting here is the fact the we can write b in function of the CDF of a non-central qui-squared random variable with 1 degree of freedom. The non-centrality parameter will just depend on the value being decreased and summed by b inside the initial probability. In this question this value is 2, so the non-centrality parameter is 4. But this can be any value you want.
The function squareroot[invF(0.4)] is the final EXACT solution for b. Also note the 0.4 can be any value between 0 and 1, not necessarily 0.4. 
