jmc

We'll consider two cases.

Case 1: $5x-3$ is nonnegative. If $5x-3$ is nonnegative, then $|5x-3|=5x-3$, so we have $5x-3\le 7$. Solving this gives $x\le 2$. The only integers for which $x\le 2$ and $5x-3$ is nonnegative are $1$ and $2$.

Case 2: $5x-3$ is negative. If $5x-3$ is negative, then $|5x-3|=-(5x-3)$, so the inequality becomes $-(5x-3)\le 7$. Multiplying by $-1$ gives $5x-3\ge -7$, so $5x\ge -4$, which means $x\ge -0.8$. The only integer greater than $-0.8$ for which $5x-3$ is negative is $0$.

Combining these cases gives us $\boxed{3}$ integers that satisfy the inequality.