For what integer $x$ is $\frac 35 < \frac x7 < \frac 79$?
For example, to check if $\frac 35 < \frac x7$, I can cross-multiply and get $21 < 5x$. Then I can divide both sides by $5$ and get $4.2 < x$. Similarly, to check if $\frac x7 < \frac 79$, I can cross-multiply and get $9x < 49$. Then I can divide both sides by $9$ and get $x < 5.44$. So, I have two inequalities: $4.2 < x$ and $x < 5.44$. I need to find an integer that satisfies both of them. I can see that the only integer between $4.2$ and $5.44$ is $5$. Therefore, $x = 5$ is the solution. 