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Systems of inequalities define conditions for variables to optimize outcomes in economics, engineering, and more. Graphical methods solve and visualize these systems, with the solution set being the intersection of all conditions. The text delves into methods for graphing and solving both two-variable and single-variable inequalities, highlighting their importance in decision-making across disciplines.

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## Definition and Applications

### Definition

Systems of inequalities are sets of two or more inequalities that work together to define the conditions that variables must satisfy

### Applications

Economics

Systems of inequalities are used in economics to determine feasible production levels, cost minimization, and resource allocation

Engineering

Systems of inequalities are used in engineering to optimize outcomes within given constraints

Operations Research

Systems of inequalities are used in operations research to find solutions that satisfy all conditions

## Graphical Methods

### Representation

Graphical methods provide a powerful means to solve and visualize systems of inequalities

### Process

To graphically solve a system of inequalities, one must first rearrange each inequality to isolate y (if necessary) and then graph the boundary of each inequality on the same coordinate plane

### Test Point Method

The test point method, typically using the origin (0,0), helps determine which side of the boundary line to shade

## Intercepts Method

### Finding Intercepts

The intercepts method is used to find the x-intercept and y-intercept of an inequality

### Graphing

The intercepts help to plot the boundary lines or curves on the graph

### Nature of Inequality

The nature of the inequality determines whether the line is solid or dashed, and the appropriate side of the boundary is shaded

## Solutions and Mastery

### Solution Set

The solution to a system of inequalities is the common area where all the regions defined by each inequality overlap

### No Solution

If no such intersection exists, the system has no solution

### Mastery

Mastery of solving systems of inequalities is essential for informed decision-making in various disciplines, as it allows for the analysis and optimization of complex situations