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## Understanding Linear Inequalities

A linear inequality is a mathematical statement that compares two linear expressions using the symbols ** (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). When graphing linear inequalities on a coordinate plane, the solution set forms a shaded region that represents all the possible values of the variables that satisfy the inequality.**

## Graphing Linear Inequalities

Graphing linear inequalities involves identifying the boundary line, determining the side of the line that represents the solution set, and shading that side on the coordinate plane. The boundary line is typically a straight line, and the shading is done based on whether the inequality represents values greater than or less than the boundary line. The solution to the inequality is then found within the shaded region.

## Understanding the Linear Inequality

The linear inequality given for graphing contains a statement comparing two linear expressions. It could be in the form of **y > mx + b**, **y , y ≥ mx + b, or y ≤ mx + b, where m represents the slope of the line and b represents the y-intercept. It’s important to interpret this inequality to identify the direction of the shading and the boundary line for accurate graphing.**

## Graphing the Linear Inequality

In order to graph the linear inequality provided, follow the steps below:

### Step 1: Identify the Boundary Line

The first step is to identify the boundary line, which is the equation of the line representing the inequality without the inequality sign. For example, if the given inequality is **y ≤ 2x + 3**, the boundary line is **y = 2x + 3**.

### Step 2: Plot the Boundary Line

Next, plot the boundary line on the coordinate plane by identifying the y-intercept at (0, b) and using the slope to plot additional points if necessary.

### Step 3: Determine the Region to Shade

After plotting the boundary line, determine which side of the line represents the solution set for the inequality. This is done by picking a test point not on the boundary line and plugging it into the original inequality. If the test point satisfies the inequality, then the region containing the test point is shaded; otherwise, the opposite region is shaded.

### Step 4: Shade the Region

Finally, shade the appropriate region on the coordinate plane based on the test point evaluation. This shaded region represents the solution set for the inequality.

## Example: Graphing a Linear Inequality

Let’s consider the linear inequality **y ≥ -2x + 4** and graph it on the provided graph below.

First, identify the boundary line:

The boundary line for the given inequality is **y = -2x + 4**.

Next, plot the boundary line on the coordinate plane:

We can identify the y-intercept at (0, 4) and use the slope to plot additional points. For example, if we move 1 unit to the right and 2 units down from the y-intercept, we plot a point at (1, 2).

Then, determine the region to shade:

To determine the region to shade, we pick a test point not on the boundary line. Let’s choose the point (0, 0). Plugging this point into the original inequality, we get:

0 ≥ -2(0) + 4

0 ≥ 4

Since this statement is false, we shade the region below the boundary line.

Finally, shade the appropriate region on the coordinate plane:

Based on the test point evaluation, we shade the region below the boundary line representing the inequality.

By following these steps, we have successfully graphed the linear inequality **y ≥ -2x + 4** on the provided graph.

## Key Points to Remember

When graphing linear inequalities, it’s important to remember the following key points:

1. Identify the boundary line, which represents the equation of the line without the inequality sign.

2. Plot the boundary line on the coordinate plane, starting with the y-intercept and using the slope to plot additional points if necessary.

3. Determine the side of the boundary line that represents the solution set by using a test point not on the boundary line.

4. Shade the appropriate region on the coordinate plane based on the test point evaluation.

## Benefits of Graphing Linear Inequalities

Graphing linear inequalities provides a visual representation of the solution set for the inequality. This allows for a better understanding of the relationships between the variables and helps in making informed decisions based on the inequality constraints. In real-world applications, such as budgeting, production planning, and resource allocation, graphing linear inequalities can aid in finding optimal solutions while considering various constraints.

## Conclusion

In conclusion, graphing linear inequalities is a valuable tool in mathematics and real-world applications. By understanding how to graph linear inequalities and accurately interpreting the solution set, individuals can make informed decisions based on the constraints represented by the inequalities. Remember to follow the steps outlined in this article and practice graphing linear inequalities to improve your skills in this area of mathematics.