MathIsimple

Systems of Inequalities & Linear Programming

Master graphing inequality systems, identify feasible regions, and apply the vertex principle for optimization in real-world decision-making scenarios!

10th Grade
Algebra
~60 min
🎮 Interactive Activity: Inequality Grapher

Identify how to graph the inequality!

Inequality:

y ≥ 2x + 1
🎮 Interactive Activity: Linear Programming Steps

Learn the steps to solve linear programming problems!

Problem: Maximize P = 3x + 2y

Subject to: x + y ≤ 10, 2x + y ≤ 16, x ≥ 0, y ≥ 0

Click below to see the solution steps

1. Introduction to Inequalities

What Are Linear Inequalities?

A linear inequality is like a linear equation, but instead of an equals sign, it uses inequality symbols: <, >, ≤, or ≥.

  • < Less than (strict inequality, use dashed line)
  • > Greater than (strict inequality, use dashed line)
  • ≤ Less than or equal to (includes boundary, use solid line)
  • ≥ Greater than or equal to (includes boundary, use solid line)
Example 1: Graphing y ≥ 2x + 1

Step 1: Graph the line y = 2x + 1

Step 2: Use a solid line (because of ≥)

Step 3: Shade above the line (y is greater than or equal to)

Test point (0, 0): 0 ≥ 2(0) + 1? No, so shade the opposite side

Example 2: Graphing x < 3

Step 1: This is a vertical line at x = 3

Step 2: Use a dashed line (because of <)

Step 3: Shade to the left (x values less than 3)

2. Graphing Linear Inequalities
3. Systems of Inequalities
4. Introduction to Linear Programming
5. The Vertex Principle
6. Real-World Applications
7. Problem-Solving Strategies
8. Advanced Topics
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
0
Correct
0%
Score
1
Which symbol requires a solid boundary line when graphing?
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2
For the inequality y > 2x + 1, which region should be shaded?
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3
What is the feasible region in linear programming?
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4
In linear programming, where does the optimal solution occur?
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5
Which inequality represents 'y is at most 5'?
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6
For the system: x + y ≤ 10, x ≥ 0, y ≥ 0, which point is in the feasible region?
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7
What does the objective function represent in linear programming?
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8
Which of the following is a constraint in linear programming?
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9
For y ≥ 3x - 2, what type of line should you graph?
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10
In the system x ≥ 0, y ≥ 0, x + y ≤ 8, 2x + y ≤ 12, which is a vertex?
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