System of Inequalities — Definition, Graph & Examples

Q: What is the difference between a system of equations and a system of inequalities?

A system of equations asks you to find exact points where all equations are true at the same time — typically a single point, a line, or no solution. A system of inequalities asks for an entire region of points that satisfy every inequality. The solution to a system of inequalities is usually a shaded area on a graph, not just one point.

Q: How do you know when to use a dashed line or a solid line?

Use a solid line when the inequality includes the boundary (≤ or ≥), because points on the line are part of the solution. Use a dashed line when the inequality is strict ( ), because points exactly on the line are not included.

System of Inequalities

Two or more inequalities containing common variable(s). Note: Systems of inequalities sometimes include equations as well as inequalities.

Example of a system of inequalities: x ≥ 0, y ≥ 0, x + y < 2

Key Formula

\begin{cases} a_1x + b_1y \leq c_1 \\ a_2x + b_2y \leq c_2 \end{cases}

Where:

$x, y$ = The common variables shared by the inequalities
$a_1, a_2$ = Coefficients of x in the first and second inequalities
$b_1, b_2$ = Coefficients of y in the first and second inequalities
$c_1, c_2$ = Constants on the right side of each inequality
$\leq$ = The inequality symbol (could also be <, >, or ≥)

Worked Example

Problem: Find and graph the solution to the system of inequalities: y ≤ 2x + 4 y > −x + 1

Step 1: Graph the boundary line for the first inequality. The equation y = 2x + 4 has a y-intercept of 4 and a slope of 2. Since the inequality uses ≤, draw a solid line.

y = 2x + 4

Step 2: Shade below the first boundary line because the inequality is y ≤ 2x + 4. You can verify by testing (0, 0): is 0 ≤ 2(0) + 4? Yes, 0 ≤ 4 is true, and (0, 0) is below the line.

0 \leq 2(0) + 4 \implies 0 \leq 4 \;\checkmark

Step 3: Graph the boundary line for the second inequality. The equation y = −x + 1 has a y-intercept of 1 and a slope of −1. Since the inequality uses > (strict), draw a dashed line.

y = -x + 1

Step 4: Shade above the second boundary line because the inequality is y > −x + 1. Test (0, 0): is 0 > −0 + 1? No, 0 > 1 is false, so (0, 0) is NOT in this region — shade the opposite side.

0 > -(0) + 1 \implies 0 > 1 \;\text{(false)}

Step 5: The solution is the overlapping region where both shadings meet. Any point in this region satisfies both inequalities. For example, test (0, 5): 5 ≤ 2(0) + 4 gives 5 ≤ 4 (false), so (0, 5) is not in the solution. Test (2, 3): 3 ≤ 2(2) + 4 gives 3 ≤ 8 (true), and 3 > −2 + 1 gives 3 > −1 (true), so (2, 3) is a solution.

\text{Point } (2,3): \quad 3 \leq 8 \;\checkmark \quad \text{and} \quad 3 > -1 \;\checkmark

Answer: The solution is the region on the coordinate plane that lies below or on the solid line y = 2x + 4 and strictly above the dashed line y = −x + 1. The point (2, 3) is one example of a solution.

Another Example

This example shows a system with three inequalities and demonstrates how to verify a specific point algebraically, without graphing. It also includes a mix of ≥, <, and ≤ symbols.

Problem: Determine whether the point (1, 2) is a solution to the system: 2x + y ≥ 4 x − y < 0 y ≤ 5

Step 1: Substitute (1, 2) into the first inequality: 2x + y ≥ 4.

2(1) + 2 = 4 \geq 4 \;\checkmark

Step 2: Substitute (1, 2) into the second inequality: x − y < 0.

1 - 2 = -1 < 0 \;\checkmark

Step 3: Substitute (1, 2) into the third inequality: y ≤ 5.

2 \leq 5 \;\checkmark

Step 4: Since (1, 2) satisfies all three inequalities, it is a solution to the system.

Answer: Yes, (1, 2) is a solution to the system because it satisfies all three inequalities.

Frequently Asked Questions

What is the difference between a system of equations and a system of inequalities?

A system of equations asks you to find exact points where all equations are true at the same time — typically a single point, a line, or no solution. A system of inequalities asks for an entire region of points that satisfy every inequality. The solution to a system of inequalities is usually a shaded area on a graph, not just one point.

How do you know when to use a dashed line or a solid line?

Use a solid line when the inequality includes the boundary (≤ or ≥), because points on the line are part of the solution. Use a dashed line when the inequality is strict (< or >), because points exactly on the line are not included.

Can a system of inequalities have no solution?

Yes. If the shaded regions of the individual inequalities do not overlap at all, the system has no solution. For example, y > 3 and y < 1 has no solution because no value of y can be greater than 3 and less than 1 at the same time.

System of Inequalities vs. System of Equations

	System of Inequalities	System of Equations
Definition	Two or more inequalities with shared variables	Two or more equations with shared variables
Solution type	A region (set of infinitely many points)	Typically a single point, a line, or no solution
Graph appearance	Overlapping shaded regions with solid or dashed boundary lines	Lines or curves that intersect at specific points
Boundary included?	Depends on ≤/≥ (solid) vs. <\> (dashed)	Points on the line are always included
Common application	Linear programming, constraints, feasibility regions	Finding exact unknown values

Why It Matters

Systems of inequalities appear frequently in algebra courses and are the foundation of linear programming, which is used to optimize real-world decisions like minimizing costs or maximizing profits. Standardized tests such as the SAT and ACT regularly include problems where you must identify a feasible region or test whether a point satisfies multiple constraints. Understanding how to graph and interpret overlapping regions builds critical skills for higher math, economics, and engineering.

Common Mistakes

Mistake: Shading the wrong side of a boundary line.

Correction: Always test a point (the origin (0, 0) is easiest if it's not on the line). Substitute its coordinates into the inequality. If the result is true, shade the side containing that point. If false, shade the opposite side.

Mistake: Using a solid line for strict inequalities (< or >).

Correction: A strict inequality means the boundary itself is NOT part of the solution. Draw a dashed line to show that points on the boundary are excluded. Reserve solid lines for ≤ and ≥.

Related Terms

Inequality — The building block of every system of inequalities
Variable — The unknown quantities shared across the system
Equation — Boundary lines come from setting each inequality as an equation
Linear Programming — Uses systems of inequalities to define feasible regions
Simultaneous Equations — The equation-based counterpart of a system of inequalities
Solution — Any point satisfying all inequalities in the system