Graphing Systems of Linear Inequalities

[EXEC: MICRO_CORE]

✖️ 1. Graphing linear inequalities on a 2D coordinate plane (solid vs. dashed boundary lines)

📏 Graphing Linear Inequalities on a 2D Coordinate Plane

Graph the boundary line by treating the inequality as an equation.
Use a solid line for $\leq$ or $\geq$ (the boundary is included).
Use a dashed line for $<$ or $>$ (the boundary is excluded).
The line divides the plane into two regions.

Example: For $y < 2x + 1$ , draw a dashed line through $y = 2x + 1$ .

💡 Solid = standing ON the line is okay; Dashed = you cannot touch the line.

[EXEC: DEEP_COMPUTE]

1. Graphing linear inequalities on a 2D coordinate plane (solid vs. dashed boundary lines)

Graphing Linear Inequalities on a 2D Coordinate Plane

A linear inequality in two variables (e.g., $ax + by \leq c$ ) divides the coordinate plane into two half-planes separated by a boundary line. The boundary line is the graph of the corresponding linear equation $ax + by = c$ .

Intuition: The boundary line acts as a threshold; points on one side satisfy the inequality, while points on the other do not.

Core Rules:

Use a solid line for $\leq$ or $\geq$ (boundary points are included in the solution set).
Use a dashed line for $<$ or $>$ (boundary points are excluded from the solution set).
Graph the boundary by finding intercepts or using slope-intercept form.
The line itself represents equality; the inequality determines which side contains solutions.

Consequence: The choice of solid versus dashed line directly affects whether boundary points satisfy the inequality.

Example: For $2x + y \geq 4$ , graph the solid line $2x + y = 4$ (intercepts at $(2, 0)$ and $(0, 4)$ ).

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["systems_of_block_1"] | DIFF_LEVEL:

[EXEC: MICRO_CORE]

✖️ 2. Shading half-planes to represent the solution set of a single inequality

🎨 Shading Half-Planes to Represent the Solution Set

Pick a test point not on the line (usually $(0, 0)$ if possible).
Substitute the test point into the inequality.
If the inequality is true, shade the side containing the test point.
If false, shade the opposite side.
All points in the shaded region satisfy the inequality.

Example: For $y > x - 3$ , test $(0, 0)$ : $0 > -3$ is true, so shade the side with $(0, 0)$ .

💡 Test point tells you which half to color in.

[EXEC: DEEP_COMPUTE]

2. Shading half-planes to represent the solution set of a single inequality

Shading Half-Planes to Represent the Solution Set

After graphing the boundary line, the solution set of a linear inequality consists of all points in one of the two half-planes created by that line. Shading visually represents this infinite set of solutions.

Intuition: Every point in the shaded region satisfies the inequality; every point outside does not.

Core Rules:

Test a point not on the boundary (commonly the origin $(0, 0)$ if not on the line) by substituting into the inequality.
If the test point satisfies the inequality, shade the half-plane containing that point.
If the test point does not satisfy the inequality, shade the opposite half-plane.
The shaded region includes the boundary line only if the inequality is $\leq$ or $\geq$ .

Consequence: The shaded half-plane represents all ordered pairs $(x, y)$ that make the inequality true.

Example: For $y < x + 1$ , test $(0, 0)$ : $0 < 1$ is true, so shade below the dashed line $y = x + 1$ .

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["systems_of_block_2"] | DIFF_LEVEL:

[EXEC: MICRO_CORE]

✖️ 3. Finding the feasible region (intersection of shaded areas) for a system of inequalities

🔍 Finding the Feasible Region

Graph all inequalities in the system on the same plane.
Shade the solution region for each inequality separately.
The feasible region is where all shaded areas overlap.
This region contains all points that satisfy every inequality simultaneously.
The feasible region can be bounded (closed polygon) or unbounded.

Example: System $x + y \leq 5$ and $x \geq 1$ creates a triangular feasible region in the first quadrant.

💡 Overlap zone = solutions that make everyone happy.

[EXEC: DEEP_COMPUTE]

3. Finding the feasible region (intersection of shaded areas) for a system of inequalities

Finding the Feasible Region

A system of linear inequalities consists of two or more inequalities considered simultaneously. The feasible region is the intersection of all individual solution sets—the area where all shaded half-planes overlap.

Intuition: Only points lying in every shaded region satisfy all inequalities at once; this overlap forms a polygonal region (possibly unbounded).

Core Rules:

Graph each inequality separately with its boundary line and shading.
Identify the overlap of all shaded regions; this is the feasible region.
The feasible region may be bounded (a polygon) or unbounded (extending infinitely).
Vertices of the feasible region occur where boundary lines intersect.

Consequence: Any point inside or on the boundary of the feasible region is a solution to the entire system; points outside violate at least one inequality.

Example: For $x + y \leq 5$ and $x \geq 1$ , the feasible region is the overlap of the half-plane below $x + y = 5$ and right of $x = 1$ .

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["systems_of_block_3"] | DIFF_LEVEL:

[EXEC: MICRO_CORE]

✖️ 4. Applications: Linear programming basics—defining feasible production regions under resource constraints in operations research

🏭 Applications: Linear Programming Basics

Linear programming finds the best outcome (maximum profit or minimum cost) under constraints.
Constraints are written as a system of linear inequalities.
The feasible region shows all possible production plans.
The optimal solution lies at a corner (vertex) of the feasible region.
Check each vertex to find which gives the best value.

Example: A factory makes chairs (x) and tables (y). Constraints: $2x + 3y \leq 120$ (labor hours), $x \geq 0$ , $y \geq 0$ . Maximize profit $P = 50x + 80y$ by testing vertices.

💡 Best answer hides at the corners of your feasible zone.

[EXEC: DEEP_COMPUTE]

4. Applications: Linear programming basics—defining feasible production regions under resource constraints in operations research

Linear Programming Basics: Feasible Production Regions

Linear programming optimizes a linear objective function (e.g., profit or cost) subject to constraints modeled as systems of linear inequalities. The feasible region represents all production plans that satisfy resource limits.

Intuition: Each inequality encodes a constraint (labor hours, materials, budget); the feasible region contains all viable production combinations.

Core Rules:

Decision variables represent quantities to determine (e.g., units of products $x$ and $y$ ).
Constraints are inequalities reflecting limited resources (e.g., $2x + 3y \leq 100$ for machine hours).
The objective function (e.g., maximize $P = 5x + 4y$ ) is evaluated at vertices of the feasible region.
The optimal solution occurs at a vertex (corner point) of the feasible region.

Consequence: Linear programming translates real-world resource allocation problems into geometric optimization over feasible regions.

Example: A factory produces chairs ( $x$ ) and tables ( $y$ ) with constraints $x + 2y \leq 40$ (wood) and $x \geq 0, y \geq 0$ ; maximize profit $P = 30x + 50y$ .

WARN: PRACTICE_BLOCK_EMPTY

QUERY_TAGS: ["systems_of_block_4"] | DIFF_LEVEL:

Systems of linear inequalities

✖️ 1. Graphing linear inequalities on a 2D coordinate plane (solid vs. dashed boundary lines)

📏 Graphing Linear Inequalities on a 2D Coordinate Plane

1. Graphing linear inequalities on a 2D coordinate plane (solid vs. dashed boundary lines)

Graphing Linear Inequalities on a 2D Coordinate Plane

WARN: PRACTICE_BLOCK_EMPTY

✖️ 2. Shading half-planes to represent the solution set of a single inequality

🎨 Shading Half-Planes to Represent the Solution Set

2. Shading half-planes to represent the solution set of a single inequality

Shading Half-Planes to Represent the Solution Set

WARN: PRACTICE_BLOCK_EMPTY

✖️ 3. Finding the feasible region (intersection of shaded areas) for a system of inequalities

🔍 Finding the Feasible Region

3. Finding the feasible region (intersection of shaded areas) for a system of inequalities

Finding the Feasible Region

WARN: PRACTICE_BLOCK_EMPTY

✖️ 4. Applications: Linear programming basics—defining feasible production regions under resource constraints in operations research

🏭 Applications: Linear Programming Basics

4. Applications: Linear programming basics—defining feasible production regions under resource constraints in operations research

Linear Programming Basics: Feasible Production Regions

WARN: PRACTICE_BLOCK_EMPTY

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