Let $$A_x = (x/2,x] \cup (x/4,x/3] \cup (x/6,x/5] \cup \cdots.$$ I can prove that $$\sum_{n \in A_x} \Lambda(n) = \log(2)x+O(\log(x)),$$ and that further, the error term cannot be improved.
Given this, an exercise in the book "A course in Analytic Number Theory" by Marius Overholt asks to deduce from this that the interval $(x/2,x]$ contains at least $x/(5\log(x))$ primes for all $x$ sufficiently large. How do I do this?