BetterGrades Precalculus · Unit 8 · Lesson

Systems of inequalities and feasible regions

Graph simultaneous inequalities, identify feasible regions, and interpret boundaries and vertices.

Opening

Start with the situation

A system of inequalities describes the intersection of several shaded solution regions.

Systems combine several conditions into one decision. Graphs, equations, inequalities, and matrices are different views of the same requirement: the final result must satisfy every condition at once.

Before you begin

Prerequisite check

Solve equations and systems.
Interpret graphs as solution sets.
Use organized arithmetic and units.

Core explanation

Explanation

Graph each boundary, shade the correct side, include contextual constraints, find vertices, and evaluate an objective when requested.

Solid boundaries are included; dashed boundaries are excluded. Feasible regions may be bounded, unbounded, or empty.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through layered half-planes, boundary inclusion, or another equivalent representation.

Conceptual reading

What the idea is really doing

A system asks for simultaneous truth. Graphs show common intersections, elimination preserves the solution set, and matrices record the same operations compactly; the representation changes, but the solution condition does not.

This lesson narrows that lens to one goal: graph simultaneous inequalities, identify feasible regions, and interpret boundaries and vertices. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Graph each boundary.
Shade the correct side.
Include contextual constraints.
Find vertices.

Verification: Substitute the result into every original equation or inequality. For matrix work, translate the final rows back into statements about variables, pivots, free variables, and consistency.

Foundation walkthrough

Plan before calculating

Problem

$x,y\ge 0, x+y\le 10, 2x+y\le 14$

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Graph each boundary, shade the correct side, include contextual constraints, find vertices, and evaluate an objective when requested.
Conclusion: Vertices $(0,0),(0,10),(4,6),(7,0)$ .
Why the check works: The constraints form a feasible polygon.

Worked examples

See the idea in three forms

foundation example

$x,y\ge 0, x+y\le 10, 2x+y\le 14$

SolutionVertices $(0,0),(0,10),(4,6),(7,0)$ .

The constraints form a feasible polygon.

representation example

Line style $y>2x+1$ .

SolutionDashed.

This example expresses systems of inequalities and feasible regions in a second form.

transfer example

Intersection $x+y=8,x+2y=10$ .

Solution $(6,2)$

Solid boundaries are included; dashed boundaries are excluded. Feasible regions may be bounded, unbounded, or empty.

Layered half-planes. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The constraints form a feasible polygon. — **Anchor figure · Layered half-planes**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: The constraints form a feasible polygon.

Boundary inclusion. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for systems of inequalities and feasible regions. — **Mechanism figure · Boundary inclusion**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for systems of inequalities and feasible regions.

Vertex objective table. Compare the valid path with the tempting shortcut. The figure shows why using the union of regions instead of their overlap leads to a false conclusion. — **Comparison and error figure · Vertex objective table**
Compare the valid path with the tempting shortcut. The figure shows why using the union of regions instead of their overlap leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is using the union of regions instead of their overlap.

Check yourself

Line style $y\le 2x+1$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Line style $y\le 2x+1$ .

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Attempt once to unlock the answer

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Practice 202

Line style $y>2x+1$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Intersection $x+y=8,x+2y=10$ .

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Complete a substantive attempt before revealing the server-held answer.

Practice 404

Define feasible point.

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Practice 505

Explain why this conclusion is valid: Vertices $(0,0),(0,10),(4,6),(7,0)$ . Use the foundation problem as evidence: $x,y\ge 0, x+y\le 10, 2x+y\le 14$ .

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Practice 606

Solve the representation example, then name the feature of systems of inequalities and feasible regions that it illustrates: Line style $y>2x+1$

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Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is using the union of regions instead of their overlap.

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Practice 808

Connect two representations for this example: $x,y\ge 0, x+y\le 10, 2x+y\le 14$ . Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: Intersection $x+y=8,x+2y=10$ . Predict the effect, solve your new example, and compare it with the original.

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Practice 1010

Write a short verification checklist for systems of inequalities and feasible regions, then apply it to one worked example from this lesson.

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Lesson close

Connect forward

The next lesson, Systems in three variables, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Utah College Algebra
Stitz and Zeager, Precalculus

No long source passage is reproduced.

x,y≥0,x+y≤10,2x+y≤14x,y\ge 0, x+y\le 10, 2x+y\le 14x,y≥0,x+y≤10,2x+y≤14

x,y≥0,x+y≤10,2x+y≤14x,y\ge 0, x+y\le 10, 2x+y\le 14x,y≥0,x+y≤10,2x+y≤14

Line style y>2x+1y>2x+1y>2x+1.

Intersection x+y=8,x+2y=10x+y=8,x+2y=10x+y=8,x+2y=10.

Original BetterGrades manuscript, rights-separated references.

$x,y\ge 0, x+y\le 10, 2x+y\le 14$

$x,y\ge 0, x+y\le 10, 2x+y\le 14$

Line style $y>2x+1$ .

Intersection $x+y=8,x+2y=10$ .