An inequality like $2x - 3y < 6$ has infinitely many solutions $(x, y)$, a whole region of the plane. Its graph combines two things you know: the graph of a line and the rules for inequalities.
Why it works
The line $2x - 3y = 6$ splits the plane into two half-planes. On one side, $2x - 3y$ is always less than $6$; on the other, always greater; and on the line it equals $6$. (Moving across the line is the only way to change which is true.) So the solutions fill exactly one side. Test any point not on the line, such as $(0, 0)$: $2(0) - 3(0) = 0 < 6$ is true, so the origin's side is shaded. The line itself is included only for $\le$ or $\ge$; for $<$ or $>$ it is drawn dashed.
Key idea
Graph the boundary line (dashed for $<$ or $>$, solid for $\le$ or $\ge$), then test a point not on the line and shade the side where the test point works.
Worked examples
Worked example: Test point method
Graph $2x - 3y < 6$.
- Boundary: $2x - 3y = 6$, with intercepts $(3, 0)$ and $(0, -2)$. Strict sign, so dashed.
- Test $(0, 0)$: $0 < 6$ is true.
- Shade the side containing the origin.
Answer: A dashed line through $(3, 0)$ and $(0, -2)$, shaded on the origin's side (above the line).
Worked example: Solving for $y$ first
Which side does $-x + 2y \ge 4$ shade?
- Add $x$: $2y \ge x + 4$. Divide by $2$ (positive, no flip): $y \ge \frac{1}{2}x + 2$.
- $y$ is greater than or equal to the line, so shade above. The sign includes equality, so the line is solid.
Answer: Solid line, shaded above.
Common mistake
Shading by the look of the sign when the inequality is not solved for $y$. In $2x - 3y < 6$, the “$<$” suggests below, but solving gives $y > \frac{2}{3}x - 2$ (dividing by $-3$ flipped it), so the solution is above. A test point settles it every time. And remember dashed for strict signs.