In mathematics, inequalities are expressions that compare two values and show their relationship using symbols like <, >, ≤, or ≥. Graphing inequalities on a coordinate plane can provide a visual representation of the solution set. By analyzing the graph, we can determine which side of the inequality includes all possible solutions. Let’s explore how to identify and interpret the inequality graphed below.

## Understanding Inequalities

Before we delve into the specific inequality graph, let’s review the basics of inequalities:

**Definition:**An inequality is a mathematical sentence that compares two values and shows their relationship using symbols like <, >, ≤, or ≥.**Types of Inequalities:**There are different types of inequalities, including linear inequalities, quadratic inequalities, rational inequalities, and more.**Solution Set:**The solution set of an inequality is the set of all values that make the inequality true.

## Interpreting the Graphed Inequality

Now, let’s examine the graph below and determine the inequality it represents:

The graph above represents a shaded region on a coordinate plane. To identify the inequality, we need to analyze the orientation of the shaded region and the line or boundary within it. Here are the steps to interpret the graphed inequality:

**Identify the boundary line:**The boundary line separates the coordinate plane into two regions. It may be a solid line (included in the solution) or a dashed line (not included in the solution).**Determine the shading:**The shading indicates which side of the boundary line includes all solutions. If the shaded region is above the line, the solutions lie above the line. If it is below the line, the solutions lie below the line.**Write the inequality:**Based on the orientation of the shaded region and the boundary line, we can write the corresponding inequality.

## Example Inequality

Let’s consider an example to illustrate how to interpret a graphed inequality:

In the example above, the boundary line is a solid line passing through the points (0, 4) and (4, 0). The shaded region is below the line. Therefore, the inequality represented by this graph is:

**y ≤ -x + 4**

Now, let’s analyze another inequality graph to determine its expression.

## Which Inequality Is Graphed Below

Below is the graph of an inequality for us to interpret:

### Analysis of the Graph

Let’s break down the elements of the graph to identify the inequality:

**Boundary Line:**The line on the graph appears to be a solid line.**Shaded Region:**The shaded region is below the line.

### Interpreting the Inequality

Based on the analysis of the graph, we can infer that the inequality represented by the graph is:

**y ≤ 2x + 3**

This inequality is in the form of y ≤ mx + b, where m represents the slope of the line and b is the y-intercept. The line has a positive slope of 2 and a y-intercept of 3, which aligns with the graphed representation.

## Conclusion

Graphing inequalities on a coordinate plane provides a visual way to represent and interpret mathematical expressions. By analyzing the orientation of the boundary line and the shaded region, we can determine the inequality represented by the graph. Understanding how to interpret inequality graphs is essential for solving mathematical problems and real-world applications.

Next time you come across a graphed inequality, follow the steps outlined in this article to identify the corresponding inequality and enhance your math skills.