Systems of Inequalities

Q: What is the solution of a system of inequalities?

The set of all points that satisfy every inequality — the region where the individual shaded half-planes overlap.

Q: How do I test whether a point is a solution?

Substitute it into each inequality. It qualifies only if it makes all of them true.

Q: Why are some boundary lines dashed?

A dashed line marks a strict inequality ($ $), meaning points exactly on the line are not included.

A system of inequalities has many solutions, not one — every point that satisfies all the inequalities at once. Graphed, that is the region where the shadings overlap.

Each inequality shades a half-plane; the solution is their intersection. Testing a point is the fast way to confirm whether it belongs.

Graphing the overlap

Graph each inequality: draw its boundary (solid for $\leq$ / $\geq$ , dashed for $<$ / $>$ ) and shade the correct half-plane. The solution region is where all the shadings coincide.

Any point in that overlap satisfies every inequality. Points in only some of the shadings do not count.

In the figure below, two half-planes are shaded with crossing hatch lines; the wedge where both hatchings overlap is the solution region — every point there satisfies both inequalities at once.

Testing a candidate point

To check whether a specific point is a solution, substitute it into every inequality. It must make all of them true.

If it fails even one, it is not in the solution region. This is faster than re-reading the graph for a single point.

Worked examples

Example 1: testing a point

Is $(1, 1)$ a solution of $y > 0$ and $y < x + 3$ ?

Check the first inequality

1 > 0 ;\checkmark

Check the second

1 < 1 + 3 = 4 ;\checkmark

Both true

\text{solution}

Answer: Yes, $(1, 1)$ is a solution

Example 2: a point that fails

Is $(0, -2)$ a solution of $y > 0$ and $y < x + 3$ ?

Check the first inequality

-2 > 0 ;\text{false}

One failure is enough

\text{not a solution}

Answer: No

Example 3: solid or dashed boundaries

Describe the boundary lines and shading for $y \geq 2x - 1$ and $y < x + 3$ .

The first allows equality, so its boundary is solid

y = 2x - 1 \text{ solid, shade above}

The second is strict, so its boundary is dashed

y = x + 3 \text{ dashed, shade below}

The solution is where both shadings coincide

\text{overlap region}

Answer: Solid line $y = 2x - 1$ shaded above, dashed line $y = x + 3$ shaded below; the solution is the overlap

Try one yourself

Common questions

What is the solution of a system of inequalities?

The set of all points that satisfy every inequality — the region where the individual shaded half-planes overlap.

How do I test whether a point is a solution?

Substitute it into each inequality. It qualifies only if it makes all of them true.

Why are some boundary lines dashed?

A dashed line marks a strict inequality ( $<$ or $>$ ), meaning points exactly on the line are not included.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Systems of Inequalities

Graphing the overlap

Testing a candidate point

Worked examples

Example 1: testing a point

Example 2: a point that fails

Example 3: solid or dashed boundaries

Try one yourself

Common questions

Keep learning

Solving Two-Variable Systems

Graphing Systems of Inequalities

Linear Programming