Graphing Systems of Inequalities

Q: When is the boundary line solid, and when is it dashed?

Solid for $\leq$ and $\geq$, because points on the line satisfy the inequality. Dashed for $ $, because points on the line make the two sides equal, and strictly less or greater rules that out.

A linear inequality like $y > 3x - 1$ does not have one line of solutions — it has a whole region of them. Its graph is a half-plane: everything on one side of a boundary line. A system of inequalities is two of these on the same graph, and the solution is where the two shaded regions overlap.

Every point in the overlap satisfies both inequalities at the same time. So instead of a single answer like $(2, 3)$ , the answer to a system of inequalities is a region — usually a wedge or strip of the plane containing infinitely many points.

Graphing one inequality: line, then shade

First draw the boundary line, which comes from replacing the inequality symbol with $=$ . Make it a solid line for $\leq$ or $\geq$ , because points on the line count as solutions. Make it a dashed line for $<$ or $>$ , because points on the line do not count.

Then shade the correct side. For $y >$ or $y \geq$ , shade above the line — those are the points with bigger $y$ -values. For $y <$ or $y \leq$ , shade below.

If you are unsure which side to shade, test a point. Pick an easy point not on the line, like $(0, 0)$ , and plug it in. If the inequality comes out true, shade the side containing that point; if false, shade the other side.

The solution is the overlap

Repeat the line-then-shade process for the second inequality on the same axes. The solution to the system is only the region where both shadings overlap — not everything you shaded, just the double-shaded part.

The graph below shows the system $y < x + 2$ and $y \geq -x + 1$ . The first boundary is dashed with shading below; the second is solid with shading above. The wedge where the regions overlap is the solution.

Checking whether a point is a solution

To decide if a specific point is in the solution, substitute it into both inequalities. It must make both true — one out of two is not enough.

Watch the boundaries. A point on a dashed boundary line is never a solution, even if it makes the other inequality true. A point on a solid boundary can be a solution, as long as it also satisfies the other inequality.

Worked examples

Example 1: graph a system

Graph the system $y < x + 2$ and $y \geq -x + 1$ .

Boundary for the first: dashed line, since

<

excludes the line

y = x + 2

Shade below it, because

y

is less than the line

Boundary for the second: solid line, since

\geq

includes the line

y = -x + 1

Shade above it, because

y

is greater than or equal to the line

The solution is the region shaded by both

Answer: The overlap of the two shaded half-planes

Example 2: test a point

Is $(0, 0)$ a solution of the system $y > x - 2$ and $y \leq -x + 2$ ?

Test the first inequality

0 > 0 - 2 \;\checkmark

Test the second inequality

0 \leq -0 + 2 \;\checkmark

Both are true, so the point is in the overlap

Answer: Yes, $(0, 0)$ is a solution

Example 3: a point that fails one inequality

Is $(4, 1)$ a solution of the system $y > x - 2$ and $y \leq -x + 2$ ?

Test the first inequality

1 > 4 - 2 \;?

1 > 2

is false, so stop — the point already fails

A solution must satisfy both inequalities

Answer: No, $(4, 1)$ is not a solution

Try one yourself

Common questions

When is the boundary line solid, and when is it dashed?

Solid for $\leq$ and $\geq$ , because points on the line satisfy the inequality. Dashed for $<$ and $>$ , because points on the line make the two sides equal, and strictly less or greater rules that out.

How do I know whether to shade above or below?

Get the inequality in the form $y$ compared to an expression. Greater means above the line, less means below. If the inequality is not solved for $y$ , either solve for $y$ first or test a point like $(0, 0)$ and shade the side that works.

What does the solution of a system of inequalities look like?

A region, not a point. Any point inside the overlap of the two shaded half-planes is a solution, so there are infinitely many. Answers to these problems are usually a graph, or a yes/no decision about specific points.

Can a system of inequalities have no solution?

Yes. If the shaded regions never overlap — for example $y > x + 3$ and $y < x - 2$ , whose parallel boundaries face away from each other — no point satisfies both, and the solution is empty.

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Graphing Systems of Inequalities

Graphing one inequality: line, then shade

The solution is the overlap

Checking whether a point is a solution

Worked examples

Example 1: graph a system

Example 2: test a point

Example 3: a point that fails one inequality

Try one yourself

Common questions

Keep learning

Writing Systems of Inequalities

Solving by Graphing

Two-Step Inequalities

Graphing Inequalities on a Number Line