We count the positive integers $x$ for which the product $(x-2)(x-4)(x-6) \cdots(x-2016)(x-2018)$ equals 0 and is less than 0 separately. The product equals 0 exactly when one of the factors equals 0. This occurs exactly when $x$ equals one of $2,4,6, \ldots, 2016,2018$. These are the even integers from 2 to 2018, inclusive, and there are $\frac{2018}{2}=1009$ such integers. The product is less than 0 exactly when none of its factors is 0 and an odd number of its factors are negative. We note further that for every integer $x$ we have $x-2>x-4>x-6>\cdots>x-2016>x-2018$. When $x=1$, we have $x-2=-1$ and so all 1009 factors are negative, making the product negative. When $x=3$, we have $x-2=1, x-4=-1$ and all of the other factors are negative, giving 1008 negative factors and so a positive product. When $x=5$, we have $x-2=3, x-4=1$ and $x-6=-1$ and all of the other factors are negative, giving 1007 negative factors and so a negative product. This pattern continues giving a negative value for the product for $x=1,5,9,13, \ldots, 2013,2017$. There are $1+\frac{2017-1}{4}=505$ such values (starting at 1, these occur every 4 integers). When $x \geq 2019$, each factor is positive and so the product is positive. Therefore, there are $1009+505=1514$ positive integers $x$ for which the product is less than or equal to 0.