Answer
We use a test point whose coordinates are not part of either equation in the system. (This means we would not use $0$ in a formula similar to $x\leq0$.)
We replace the variable(s) in the first equation and check to see if the equation is true or false. If the equation is true, then we consider shading the side of the line with the test point. If the equation is false, then we consider shading the side of the line without the test point.
This process is repeated for the second equation (and any subsequent equations). After the process is repeated for all equations, we find the area that all equations had in common (and were true).
Work Step by Step
Example:
For the two lines in this system, we can use the origin $(0,0)$ as our test point.
$x\geq3$ is the first inequality.
$x\geq 3$
$0\geq 3$ is a false statement, so we would consider shading the side of the line without the origin.
$y\geq-2$ is the second inequality.
$y\geq-2$
$0\geq-2$ is a true statement, so we would consider shading the side of the line with the origin.
In the screenshot, the solution region is the area shaded purple. (The blue region is for the line $x\ge3$, and the red region is for the line $y\ge-2$.)
