How many positive integer solutions exist for $3(x-5)\le 7$ ?
This problem is asking for the number of values of x that satisfy the given inequality. I want to isolate x on one side of the inequality, so I will start by expanding the parentheses and simplifying. I get $3x-15\le 7$, which I can add 15 to both sides to get $3x\le 22$. Now I want to divide both sides by 3, but I have to be careful with the direction of the inequality sign. Since 3 is positive, dividing by 3 will not change the direction of the inequality, so I get $x\le \frac{22}{3}$. Now I have to think about what this means in terms of positive integer solutions. I know that $\frac{22}{3}$ is not an integer, but it is between two consecutive integers: 7 and 8. That means that the largest integer value of x that satisfies the inequality is 7. I also know that x has to be positive, so the smallest integer value of x that satisfies the inequality is 1. Therefore, the possible integer values of x are 1, 2, 3, 4, 5, 6, and 7. There are 7 of them, so that is the answer. 