# How to Graph Solutions of System of Inequalities Using Julius AI

In this detailed guide, I’m going to help you explore how to interact with Julius AI to solve mathematical problems, specifically focusing on finding the corners of the feasible region defined by a system of inequalities and graphing the solution of another system of inequalities. This walkthrough will help you understand how to formulate your queries and interpret the responses from Julius AI effectively.

## (Optional) Finding the Corners of the Feasible Region

### Step 1: Formulate the System of Inequalities

Start by clearly defining your system of inequalities. For example, consider a system like:

```
3x + 12y ≤ 180
6x + 3y ≤ 150
x ≥ 0
y ≥ 0
```

Your goal is to find the corners of the feasible region defined by these inequalities.

### Step 2: Convert Inequalities to Boundary Equations

Ask Julius AI to convert the inequalities into boundary equations by replacing the inequality signs with equal signs. This step is crucial for identifying the exact lines that form the boundaries of your feasible region.

### Step 3: Determine the Intersections

Request Julius AI to find the intersections of these boundary equations. The intersections will be the corners of your feasible region. Julius AI will solve the equations in pairs to find these points, considering only the solutions that fall in the first quadrant due to the non-negativity constraints (x ≥ 0, y ≥ 0).

### Step 4: Interpret the Results

Julius AI will provide you with the coordinates of the intersections. These points are the corners of the feasible region for your system of inequalities. Make sure to understand how each intersection corresponds to the equations involved.

## Graphing the Solution of a System of Inequalities

### Step 1: Define Your System of Inequalities

Consider a new system of inequalities for graphing, such as:

```
y < 3x
y > x - 2
```

Your objective is to graph the solution of this system.

### Step 2: Request a Graph from Julius AI

Ask Julius AI to graph the solution of your system of inequalities. Be specific about your request, mentioning that you want to see the areas where the inequalities overlap if applicable.

### Step 3: Julius AI Processes Your Request

Julius AI will use mathematical and graphing libraries (like NumPy and Matplotlib in Python) to create a visual representation of the solution. It will define a range of x values, plot the inequalities, and fill the region that satisfies both inequalities.

### Step 4: View the Graph

Julius AI will provide a link to the graph or directly show the graph in the chat. Examine the graph to understand the solution region of your system of inequalities. The shaded area represents the set of points that satisfy all inequalities in the system.

## Conclusion

By following these steps, you can effectively use Julius AI to find the corners of the feasible region for a system of inequalities and to graph the solution of another system. Remember to clearly define your problem, understand the steps Julius AI takes to solve it, and interpret the results or graphs provided. This guide should serve as a helpful resource for tackling similar mathematical problems using Julius AI.

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