Graph the linear inequality y >= 0.

Step-by-step solution

1. 1

   Read what the inequality does and does not restrict. There is no $x$ anywhere in $y\ge 0$. That means $x$ is completely free: whatever restriction exists applies to the height of a point only. Any horizontal line you draw is either entirely in the solution set or entirely out of it.

2. 2

   Find the boundary by replacing the inequality with an equation.

   $y=0$

   That is the $x$-axis. It splits the plane into the region above it and the region below it.

3. 3

   Decide solid or dashed. The symbol is $\ge$, not $>$, so points on the boundary satisfy the inequality ($0\ge 0$ is true). Draw the $x$-axis as a solid line. A dashed line would wrongly exclude every point of the form $(x,0)$.

4. 4

   Pick a test point off the boundary. Take $(0,1)$: substituting gives $1\ge 0$, which is true, so the half-plane containing $(0,1)$ — the one above the $x$-axis — is shaded. Testing $(0,-1)$ gives $-1\ge 0$, false, confirming the lower half-plane is excluded.

5. 5

   Describe the solution set. In set-builder form,

   $\{(x,y)\in\mathbb{R}^2 : y\ge 0\}$

   the closed upper half-plane: every point on or above the $x$-axis, with $x$ unrestricted. This is the first quadrant together with the second quadrant, the whole $x$-axis, and the positive part of the $y$-axis.