Solve the compound inequality.
$$-5x\le -15\text{ or }6x-23\ge 19$$
Select the correct choice below and, if nece

Step 1 ：The given inequality is a compound inequality. It can be solved by isolating the variable 'x' in each inequality.

Step 2 ：For the first inequality, we can isolate 'x' by dividing all parts by -5. Remember that when we divide or multiply an inequality by a negative number, the inequality sign flips. So, $-5x\le -15$ becomes $x\ge 3$.

Step 3 ：For the second inequality, we can isolate 'x' by first adding 23 to both sides, and then dividing by 6. So, $6x-23\ge 19$ becomes $x\ge 7$.

Step 4 ：Since the original inequality is an 'or' inequality, the solution is the union of the solutions to the two inequalities. So, the solution to the compound inequality is $x\ge 3$ or $x\ge 7$.

Step 5 ：Final Answer: The solution to the compound inequality is $x\ge 3$ or $x\ge 7$. This can be written in interval notation as $\boxed{{\displaystyle [3,\mathrm{\infty })\cup [7,\mathrm{\infty })}}$. However, since [7, \infty) is a subset of [3, \infty), we can simplify this to just $\boxed{{\displaystyle [3,\mathrm{\infty })}}$.