Lesson 7.15

Systems of Inequalities

What happens when we impose two rules at once? "You must be taller than 4ft AND weigh less than 100lbs." The solution is the area where BOTH rules are obeyed.

Introduction

A System of Inequalities works just like a system of equations, but instead of finding a point , we are finding a Region.

Past Knowledge

Lesson 7.13 (Inequalities). You need to be fast at graphing single inequalities.

Today's Goal

Identify the solution region by finding the overlap of two shaded areas.

Future Success

This is the final step of Unit 7. The visual "Feasible Region" is the key to optimization problems.

Key Concepts

The Overlap Zone

The solution is ONLY the part where the shading overlaps.

Shade A
+
Shade B
=
SOLUTION

The Darkest Part

Any point in the purple zone satisfies BOTH rules.

Worked Examples

Example 1: The "V" Shape

Basic

Graph the system:

  • (Blue)
  • (Red)

Analysis

1. Graph dashed. Shade Above.

2. Graph dashed. Shade Below.

3. Highlight the overlapping wedge.

Example 2: A Bounded Strip

Intermediate

Graph:

Analysis

1. Horizontal line at . Shade Below.

2. Horizontal line at . Shade Above.

Result: The strip between the two lines.

Example 3: No Solution

Advanced

Graph:

Analysis

Parallel lines.

Eq 1: Shade Above top line.

Eq 2: Shade Below bottom line.

They NEVER touch. No Solution.

Common Pitfalls

Only Shading One

You must find the OVERLAP. Shading just one color is only doing half the problem. The final answer is the region with double-shading.

Testing the Solution

Students often pick a point on a DASHED boundary line. Points on the dashed line are NOT solutions! Only points safely inside the region count.

Real-Life Applications

Civil Engineering:

  • "The road must be at least 20 feet wide () but stay within the 50 foot easement ()."
  • Combining constraints helps engineers design safe and legal structures.

Practice Quiz

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