Resulting expected value of a bounded exponential pdf

#1
I need to determine the distribution function of an exponential random variable with mean 2, given that its outcome is larger than 2 and give the resulting expected value?

I know that the mean is 1/lambda, so lambda should be 1/2.

I get that way 0.5*e^(-0.5x) >= 2. How can I deal with the inequality if I want to compute the expected value ?