Choice C is correct. All the tables in the choices have the same three values of $x$, so each of the three values of $x$ can be substituted in the given inequality to compare the corresponding values of $y$ in each of the tables. Substituting $3$ for $x$ in the given inequality yields $y<6\left(3\right)+2$, or $y<20$. Therefore, when $x=3$, the corresponding value of $y$ is less than $20$. Substituting $5$ for $x$ in the given inequality yields $y<6\left(5\right)+2$, or $y<32$. Therefore, when $x=5$, the corresponding value of $y$ is less than $32$. Substituting $7$ for $x$ in the given inequality yields $y<6\left(7\right)+2$, or $y<44$. Therefore, when $x=7$, the corresponding value of $y$ is less than $44$. For the table in choice C, when $x=3$, the corresponding value of $y$ is $16$, which is less than $20$; when $x=5$, the corresponding value of $y$ is $28$, which is less than $32$; when $x=7$, the corresponding value of $y$ is $40$, which is less than $44$. Therefore, the table in choice C gives values of $x$ and their corresponding values of $y$ that are all solutions to the given inequality.

Choice A is incorrect. In the table for choice A, when $x=3$, the corresponding value of $y$ is $20$, which is not less than $20$; when $x=5$, the corresponding value of $y$ is $32$, which is not less than $32$; when $x=7$, the corresponding value of $y$ is $44$, which is not less than $44$.

Choice B is incorrect. In the table for choice B, when $x=5$, the corresponding value of $y$ is $36$, which is not less than $32$.

Choice D is incorrect. In the table for choice D, when $x=3$, the corresponding value of $y$ is $24$, which is not less than $20$; when $x=5$, the corresponding value of $y$ is $36$, which is not less than $32$; when $x=7$, the corresponding value of $y$ is $48$, which is not less than $44$.