A linear inequality in two variables describes a whole region of the plane, not just a line. This subtopic covers such inequalities, shading regions, and dashed vs solid boundary lines.
What is it?
A linear inequality in two variables has the form $y > mx+c$, $y \geq mx+c$, $y < mx+c$ or $y \leq mx+c$ (and similar forms with $x$). Its solution is a region of the coordinate plane.
Key formula
$y > mx+c \quad\text{or}\quad y < mx+c$
Dashed vs solid line
First draw the boundary line $y = mx+c$.
- Use a solid line for $\geq$ or $\leq$ — points on the line are included.
- Use a dashed line for $>$ or $<$ — points on the line are not included.
Shading the region
For $y > \ldots$ shade above the line; for $y < \ldots$ shade below. To be sure, pick a test point such as $(0,0)$: if it satisfies the inequality, shade its side.
Example
Test $(0,0)$ in $y > 2x+1$: $0 > 1$ is false, so shade the side not containing the origin, using a dashed line.
Remember
Dashed = strict ($>,<$); solid = inclusive ($\geq,\leq$). Always confirm the region with a test point.