Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 1 systems of inequalities word problems? Then this Algebra 1 lesson on Graphing Inequalities and Systems with Shaded Regions explains each idea step by step, with worked examples.
Quick answer: The graph of a linear inequality is a half-plane bounded by a line: solid for ≤ or ≥ and dashed for < or >. Shade the side that contains true test points. A system of inequalities is graphed by shading each half-plane, and its solutions are the points in the overlapping region.
Standards: A-REI.D.12, A-CED.A.3 · Level: Grade 9 Algebra 1
How do you graph a linear inequality?
The graph of a linear inequality is a half-plane: every point on one side of a boundary line. Three steps produce it.
- Graph the boundary y = −x + 4. Use a dashed line for < or > and a solid line for ≤ or ≥.
- Test a point that is not on the line. For (0, 0): 0 ≤ 0 + 4 is true.
- Shade the side containing the true test point, here the region below the line.
A strict inequality leaves the boundary out. For y > 2x − 1 the line is dashed. The test point (0, 0) gives 0 > −1, which is true, so the origin is on the shaded side, above the line.
- ≤ or ≥: solid line, boundary points are solutions.
- < or >: dashed line, boundary points are not solutions.
What is a system of inequalities?
A system of inequalities is two or more inequalities that must all be true. Graph each half-plane, and the solution region is where the shadings overlap.
| Step | What to do | In the graph above |
|---|---|---|
| 1 | graph each boundary | dashed y = x − 3, solid y = −x + 5 |
| 2 | shade each half-plane | above the first line, below the second |
| 3 | mark the overlap and test a point | (4, 1) is on the dashed line, so it is not a solution |
How do constraints become inequalities?
Words translate into symbols: “at most” is ≤, “at least” is ≥, and “between” needs two inequalities. In real problems the variables often cannot be negative, adding x ≥ 0 and y ≥ 0. A point of the overlap is a feasible choice, and its corners come from solving pairs of boundary equations.
Using a solid line for a strict inequality, or forgetting to reverse the sign when dividing by a negative number while solving for y.
Always test a point from the shaded region in every original inequality before you finish.
- A boundary line is solid for ≤ and ≥, dashed for < and >.
- Test a point not on the line to decide which side to shade.
- Dividing by a negative number reverses the inequality sign.
- The solution of a system is the overlap of all the shaded half-planes.
- In real problems, add x ≥ 0 and y ≥ 0 when quantities cannot be negative.
Graphing Inequalities and Systems with Shaded Regions: frequently asked questions
When is the boundary line dashed?
The line is dashed when the inequality is strict (< or >), because points on the line do not make the inequality true. It is solid for ≤ and ≥.
How do you choose which side to shade?
Pick a test point not on the line, often (0, 0), and substitute it. If the statement is true, shade the side with that point; if false, shade the other side.
Why does dividing by a negative number reverse the sign?
Multiplying or dividing by a negative number reverses the order of values. In 4x − 2y ≤ 10, dividing by −2 gives y ≥ 2x − 5, so the shading is above the line.
What is a feasible region?
It is the set of points that satisfy every constraint in a real problem. Its corner points come from the intersections of the boundary lines and often help find a best choice.
Can a system of inequalities have no solution?
Yes. If the half-planes do not overlap, as with y ≥ 2x + 1 and y ≤ 2x − 3, no point satisfies both and the system has no solution.
Mis à jour le