The number of integers x satisfying (x-1)/(x-4)≤0 is

The fraction is non-positive when $x$ lies between the zero of the numerator and the zero of the denominator. Hence $1\le x<4$: the endpoint $x=1$ is allowed (value $0$) but $x=4$ is excluded because the expression is undefined. The integers are $1,2,3$, giving $3$. Cross-multiplying without checking signs can wrongly include $4$.