Question

How do you graph the system of linear inequalities `2x+1>=y` and `x<5` and `y<x+2`?

Answer 1

Draw solid or dashed lines corresponding to equations, then test the origin, `(0,0)`, to shade. Where all three overlap is the final answer.

Explanation:

One way to graph a system of linear equations like this is to actually start by drawing them as if they were equalities first.

The inequality `2x+1 >= y` becomes `y=2x+1` When you graph that, you get
![Desmos.com and MS Paint]()

Testing the point `(0,0)`, you see that it's true that `1 >= 0`, so shade that side. Because the inequality is "less then or equal to", you draw a solid (not dashed) line.
![Desmos.com and MS Paint]()

Again, assume `x < 5` is really just `x=5`. This gives:
![Desmos.com and MS Paint]()

This time, the graph is dashed because you were given `x < 5`. Testing the point `(0,0)`, it is true that `x=0` is less than `5`, so we shade to the left.

Finally, pretending to graph `y < x+2` gives the line. We must used a dashed line and shade wherever the inequality is true. At the point, `(0,0)`, the inequality is false, which means we must shade the other side.
![Desmos.com and MS Paint]()

Finally, if we are to find out where ALL THREE inequality are true, you simply look for the solution where ALL THREE shaded parts overlap. That occurs here:
![Desmos.com and MS Paint]()