**Which of the Following Inequalities Matches the Graph? Unveiling the Equation Behind the Curve**

In the realm of mathematics, graphs and inequalities intertwine to create a tapestry of visual representations and mathematical relationships. Delving into this fascinating interplay, we embark on a journey to uncover which inequality matches a given graph, unraveling the equation that governs the curve.

Inequalities, the guardians of mathematical boundaries, dictate the permissible values for variables, shaping the contours of graphs. They delineate regions where a function exists, creating patterns and structures that intrigue the mathematical eye. These inequalities, expressed through symbols such as <, >, ≤, and ≥, serve as gatekeepers, determining which points lie within the realm of possibility and which fall beyond its grasp.

To determine which inequality matches a given graph, we embark on a detective-like investigation, scrutinizing the curve’s characteristics. We examine its shape, identifying whether it rises or falls, and whether it possesses any asymptotes, those elusive lines that the curve approaches but never intersects. We also take note of any intercepts, the points where the graph intersects the coordinate axes.

Armed with these observations, we construct an inequality that aligns with the graph’s behavior. We consider various options, testing each one against the graph, until we find the equation that perfectly captures the curve’s essence. This inequality becomes the mathematical doppelgänger of the graph, a symbolic representation of its visual manifestation.

In conclusion, determining which inequality matches a graph is an exercise in visual analysis and mathematical deduction. It requires a keen eye for patterns, an understanding of inequalities’ properties, and the ability to translate graphical characteristics into mathematical expressions. By embracing this challenge, we not only unveil the equation behind the curve but also deepen our appreciation for the intricate relationship between graphs and inequalities.

**Which of the Following Inequalities Matches the Graph?**

**Introduction:**

Inequalities are mathematical statements that compare two expressions, using symbols such as <, >, ≤, or ≥. Inequalities can be represented graphically, with the solution region shaded to indicate the values that satisfy the inequality. This article aims to determine the inequality that corresponds to a given graph.

**Understanding the Graph:**

To determine the inequality that matches the graph, we need to understand the concept of slope and y-intercept. Slope is the steepness of a line, while y-intercept is the point where the line crosses the y-axis.

**1. Slope:**

The slope of a line can be positive or negative. A positive slope indicates that the line rises from left to right, while a negative slope indicates that the line falls from left to right. The slope of the graph can be determined using the following formula:

```
Slope = (y2 - y1) / (x2 - x1)
```

Where (x1, y1) and (x2, y2) are two points on the line.

**2. Y-Intercept:**

The y-intercept of a line is the point where the line crosses the y-axis. The y-intercept can be determined by substituting x = 0 into the equation of the line.

**Analyzing the Graph:**

By analyzing the given graph, we can observe the following characteristics:

**1. Slope:**

The slope of the line is positive, indicating that the line rises from left to right.

**2. Y-Intercept:**

The line crosses the y-axis at the point (0, 2).

**3. Shaded Region:**

The shaded region is below the line.

**Identifying the Inequality:**

Based on the characteristics of the graph, we can identify the inequality that matches the graph as follows:

**1. Linear Inequality:**

The inequality is a linear inequality, which is an inequality that can be represented by a straight line.

**2. Type of Inequality:**

Since the shaded region is below the line, the inequality is of the form y < mx + b, where m is the slope and b is the y-intercept.

**3. Determining the Coefficients:**

Substituting the slope (m = 1) and y-intercept (b = 2) into the inequality, we get:

```
y < x + 2
```

**Conclusion:**

Therefore, the inequality that matches the given graph is y < x + 2. This inequality represents the shaded region below the line with a positive slope and a y-intercept of 2.

**FAQs:**

**1. What is the slope of the line in the graph?**

Answer: The slope of the line is 1, indicating a positive slope.

**2. What is the y-intercept of the line in the graph?**

Answer: The y-intercept of the line is 2, indicating that the line crosses the y-axis at the point (0, 2).

**3. What is the type of inequality that matches the graph?**

Answer: The inequality that matches the graph is a linear inequality of the form y < mx + b, where m is the slope and b is the y-intercept.

**4. What is the inequality that matches the graph?**

Answer: The inequality that matches the graph is y < x + 2.

**5. What does the shaded region in the graph represent?**

Answer: The shaded region in the graph represents the values of y that satisfy the inequality y < x + 2.

.

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