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

Jun 25, 2017

Draw solid or dashed lines corresponding to equations, then test the origin, $\left(0 , 0\right)$, 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 $2 x + 1 \ge y$ becomes $y = 2 x + 1$ When you graph that, you get

Testing the point $\left(0 , 0\right)$, you see that it's true that $1 \ge 0$, so shade that side. Because the inequality is "less then or equal to", you draw a solid (not dashed) line.

Again, assume $x < 5$ is really just $x = 5$. This gives:

This time, the graph is dashed because you were given $x < 5$. Testing the point $\left(0 , 0\right)$, 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, $\left(0 , 0\right)$, the inequality is false, which means we must shade the other side.

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: