A system of inequalities asks for points that satisfy every inequality at the same time. Graph each condition, then identify the overlap of all shaded regions.
Example system
y ≥ x^2 y ≤ 2x + 3
The solution lies above the parabola and below the line.
Find the boundary intersections
Solve x^2=2x+3. The intersections occur at x=-1 and x=3. The common region exists between these x-values.
Draw each boundary correctly
Both inequalities include equality, so both boundaries are solid. If either relation used < or >, that boundary would be dashed.
Shade one inequality at a time
- Shade on or above
y=x^2. - Shade on or below
y=2x+3. - Keep the region where the two shadings overlap.
Use contrasting colors
Different semi-transparent colors make the overlap easy to see. Avoid full opacity because it can hide boundaries and other shaded regions.
A linear example
x + y ≤ 4 x ≥ 0 y ≥ 0
The solution is the triangular region in the first quadrant bounded by the coordinate axes and the line x+y=4.
Check a point from the overlap
Select a point visibly inside the proposed solution region and substitute it into every inequality. A point is part of the solution only if all statements are true.
Empty and unbounded solutions
A system may have no common region, or the overlap may extend indefinitely. Adjust the viewing window before deciding that a solution is empty or bounded.
Create the system online
Use Equation / Inequality objects for direct relations or combine boundary functions with multiple regions in the Graph Shading tool. Edit colors, opacity, and patterns so the final overlap remains readable.
Common mistakes
- Taking the union of regions when the problem requires their intersection.
- Using the wrong boundary style.
- Forgetting a condition such as
x≥0. - Testing a point against only one inequality.
- Using colors that make the overlap indistinguishable.
Graph the example system and inspect the overlap →
When a shaded plane region is rotated, it becomes a three-dimensional object. Learn how in the solid-of-revolution guide.