5.1Introduction to Systems of Inequalities
A system of inequalities is two or more inequalities with the same variables. The solution is the region of the coordinate plane where all inequalities are satisfied simultaneously.
Why This Matters
Systems of inequalities define feasible regions — the set of all solutions that satisfy multiple constraints at once. This is the mathematical foundation of linear programming, used in logistics, manufacturing, and operations research.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
What does it mean for an ordered pair to be a solution to a system of inequalities, and how do we represent all solutions graphically?
Lesson Overview
A system of inequalities is a set of two or more inequalities that share the same variables. Unlike a system of equations — which has a single point as its solution — a system of inequalities has a solution region (also called the feasible region): the set of all ordered pairs that satisfy every inequality at the same time. To graph a system, you graph each inequality individually and then identify the overlapping shaded region.
Steps to Graph a System of Inequalities
- Graph the boundary line for each inequality. Use a solid line for ≤ or ≥; use a dashed line for < or >.
- Choose a test point (usually (0, 0) if it is not on the line) and substitute it into the inequality.
- Shade the correct half-plane: if the test point satisfies the inequality, shade the side containing it; otherwise, shade the opposite side.
- Repeat for each inequality.
- Identify the feasible region: the area where all shaded regions overlap is the solution set.
y ≥ x: solid boundary, shade above
y < −x + 3: dashed boundary, shade below
Worked Examples
Solid vs. dashed boundary lines
Feasible region: overlap of y ≥ x and y < −x + 3
Is (2, 3) a solution to the system: y > x + 1 and y ≤ 4?
Check Inequality 1: y > x + 1 → 3 > 2 + 1 = 3 → 3 > 3 is FALSE.
Since (2, 3) fails Inequality 1, it is NOT a solution.
Is (1, 5) a solution to the system: y ≥ 2x + 1 and y < 3x + 4?
Check Inequality 1: y ≥ 2x + 1 → 5 ≥ 2(1) + 1 = 3 → 5 ≥ 3 ✓
Check Inequality 2: y < 3x + 4 → 5 < 3(1) + 4 = 7 → 5 < 7 ✓
Both inequalities are satisfied.
Describe how to graph the system: y ≤ x + 2 and y > −x + 1.
Graph y = x + 2 as a SOLID line (≤ includes the boundary). Shade below (y ≤ means below the line).
Graph y = −x + 1 as a DASHED line (> does not include the boundary). Shade above (y > means above the line).
The feasible region is where both shaded areas overlap.
Classify the boundary lines for: y ≥ 3x − 1 and y < −2x + 5.
y ≥ 3x − 1: the symbol is ≥ → SOLID boundary line; shade above (y ≥ means above or on the line).
y < −2x + 5: the symbol is < → DASHED boundary line; shade below (y < means below the line).
Is (0, 0) a solution to the system: y ≥ x − 3 and y ≤ −x + 4?
Check Inequality 1: 0 ≥ 0 − 3 = −3 → 0 ≥ −3 ✓
Check Inequality 2: 0 ≤ −0 + 4 = 4 → 0 ≤ 4 ✓
Both inequalities are satisfied.
Guided Practice
Guided Practice Video: Introduction to Systems of Inequalities
Watch the guided practice walkthrough for introduction to systems of inequalities, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Is (3, 4) a solution to the system y ≥ x + 1 and y < 2x − 1? Substitute and verify both inequalities.
Hint: Substitute (3, 4) into each inequality separately. Both must be true for it to be a solution.
Is (0, 0) a solution to the system y > −x + 2 and y ≤ 3x + 1?
Hint: Substitute (0, 0) into both inequalities. Remember: 0 > 2 is false.
For the inequality y ≤ 2x − 3, should the boundary line be solid or dashed? Which side should be shaded?
Hint: The symbol ≤ includes equality, so the boundary is solid. Since y ≤ means y is less than or equal to, shade below the line.
For the inequality y > −x + 4, should the boundary line be solid or dashed? Which side should be shaded?
Hint: The symbol > is strict (no equality), so the boundary is dashed. Shade above the line.
Describe the feasible region for the system: y ≥ 0 and x ≥ 0 and y ≤ 4.
Hint: Graph all three inequalities. The feasible region is bounded by the x-axis, y-axis, and the horizontal line y = 4.
Test Point Method
To determine which side to shade, pick a test point NOT on the boundary line (usually (0,0)).
Inequality: y > 2x − 1
Test (0,0): 0 > 2(0) − 1 → 0 > −1 ✓ TRUE
→ Shade the side containing (0,0)
Key Vocabulary
System of Inequalities
Two or more inequalities with the same variables considered together.
Example: y > x + 1 and y ≤ 4 form a system.
Solution of a System
An ordered pair (x, y) that satisfies every inequality in the system simultaneously.
Example: (1, 5) satisfies both y ≥ 2x + 1 and y < 3x + 4.
Feasible Region
The overlapping shaded area on a graph that represents all solutions to a system of inequalities.
Example: The region where all shaded half-planes overlap.
Boundary Line
The line that forms the edge of the solution region. Solid (≤ or ≥) if the line is included; dashed (< or >) if not.
Example: y ≤ x + 2 → solid line; y > x + 2 → dashed line.
Half-Plane
The region on one side of a boundary line. Each inequality divides the plane into two half-planes.
Example: y > 3 is the half-plane above the line y = 3.
Test Point
A point (often the origin) substituted into an inequality to determine which half-plane to shade.
Example: Test (0, 0) in y > x + 1: 0 > 1 is false → shade the other side.
Interactive Practice — 5 Questions
Which ordered pair is a solution to the system y > x + 1 and y ≤ 3x − 2?
For the inequality y ≥ 2x − 5, the boundary line should be:
The feasible region of a system of inequalities represents:
Is (0, 0) a solution to y < x + 3 and y ≥ −2x − 1?
A point lies on the dashed boundary line of y > 3x − 2. Is it a solution?
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Is (2, 5) a solution to y ≥ x + 3 and y < 2x + 2? Verify both inequalities.
For y < 3x − 1: solid or dashed boundary? Shade above or below?
For y ≥ −2x + 4: solid or dashed boundary? Shade above or below?
Is (5, 5) a solution to y ≤ x and y ≥ −x + 4? Show your work.
A point is on the solid boundary line of y ≥ x + 3. Is it a solution? Explain.
Common Mistakes
Treating the solution region like a single point — writing one ordered pair as the answer.
A system of inequalities has infinitely many solutions. The answer is the entire shaded feasible region.
Using a solid boundary line for a strict inequality (< or >).
Strict inequalities (< or >) use a dashed boundary line. Use a solid line only for ≤ or ≥.
Shading the wrong side of the boundary line — shading above when the solution is below.
Test a point (like the origin) in the inequality. If it satisfies the inequality, shade that side.
Forgetting to shade the overlap — shading each inequality separately without finding the intersection.
The feasible region is where both (or all) shadings overlap. Only that overlapping area is the solution.
Math Tips
Solid vs. dashed: ≤ and ≥ use a solid boundary line (the line is part of the solution); < and > use a dashed line (the line is NOT part of the solution).
Test point shortcut: (0, 0) works as a test point unless the boundary line passes through the origin — then use (1, 0) or (0, 1).
Checking a solution: substitute the ordered pair into every inequality — it must satisfy all of them.
The feasible region can be bounded (a closed polygon) or unbounded (extends to infinity).
Points on a dashed boundary line are NOT solutions; points on a solid boundary line ARE solutions.