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System of Inequalities

Solve systems of linear inequalities. Feasible region, vertices, bounded/unbounded. Linear programming setup.

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Why: Understanding system of inequalities helps you make better, data-driven decisions.

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📐 Examples — Click to Load

Inequalities: ax + by op c

inequalities.sh
CALCULATED
$ solve --inequalities
Feasible
Unbounded region
Vertices
1
Bounded
No
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Boundary Intercepts

Feasible vs Infeasible

📐 Steps

Step 1: System of inequalities
1. 1x + 1y ≤ 5
2. 1x - 1y ≥ 0
Step 2: Boundary lines (ax + by = c)
Line 1: 1x + 1y = 5
Line 2: 1x + -1y = 0
Step 3: Find feasible vertices (intersections satisfying all)
Vertex 1: (2.5, 2.5)
Step 4: Feasible region: Unbounded region

For educational and informational purposes only. Verify with a qualified professional.

📋 Key Takeaways

  • • Feasible region: Set of points satisfying all inequalities
  • • Graphical method: Graph each boundary line, shade the half-plane, find overlap
  • • Corner point theorem: For LP, optimum lies at a vertex of the feasible region
  • • Bounded vs unbounded: Bounded = finite area; unbounded = extends infinitely
  • • Linear programming: Optimize a linear function over the feasible region

📖 Graphical Method

  1. Graph each inequality as a half-plane (solid line for ≤/≥, dashed for </>)
  2. Shade the side that satisfies the inequality (test (0,0) if on that side)
  3. The feasible region is the intersection of all shaded regions
  4. Find vertices by solving boundary line intersections

❓ FAQ

What is the feasible region?

The set of all points (x,y) that satisfy every inequality in the system simultaneously.

When is the region bounded?

When it forms a closed polygon (finite area). Typically need at least 3 constraints that "close" the region.

What is the corner point theorem?

For linear programming, if an optimal solution exists, it occurs at a vertex (corner) of the feasible region.

How do I test a point?

Plug (x,y) into each inequality. If all are true, the point is feasible.

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