I believe you are right. The phrase "the right hand side is at the minimum when x + 3 is at its maximum" shows that they found the minimum of e / |x + 3|. But in order to bound this expression from above, they should have found its maximum, which is achieved when |x + 3| is at its minimum, i.e., 4. Therefore, e / |x + 3| < e / 4, not e / 6. Good catch.