**Unlocking the Key to Compound Inequalities: Which Graph Reveals the Solution Set?**

Navigating compound inequalities can be a perplexing maze, leaving you wondering which path leads to the right solution. In this guide, we’ll shed light on this enigma by unraveling the mysteries of graph representations.

**Unveiling the Perils of Uncharted Territories**

When confronted with compound inequalities, many struggle to visualize the intricate boundaries that define the solution set. This uncertainty can lead to frustration and wasted efforts if not addressed effectively.

**Illuminating the Path to Clarity**

The key to unraveling the enigma of compound inequalities lies in understanding how to represent their solutions graphically. By plotting the values that satisfy both inequalities, we can create a shaded region that represents the solution set. Determining which graph accurately depicts this region is crucial for solving the inequality correctly.

**Unleashing the Power of Graphs**

In the vast realm of graphs, there are two primary types that can represent the solution set of a compound inequality: open and closed intervals. Open intervals are unbounded lines, represented by the notation (a, b), where a and b are the endpoints, and points on the lines are not included in the solution set. Closed intervals, on the other hand, are solid lines, denoted by [a, b], and include the points on the endpoints. By identifying the correct interval representation, you can accurately determine the solution set of the compound inequality.

## How to Graph the Solution Set of a Compound Inequality

In mathematics, a compound inequality is an inequality that contains two or more inequalities joined by the word “and” or “or.” To graph the solution set of a compound inequality, you need to first graph the solution set of each individual inequality. Then, you can combine the solution sets to find the solution set of the compound inequality.

### Graphing the Solution Set of an Inequality

To graph the solution set of an inequality, you need to first determine the critical points of the inequality. The critical points are the values of the variable that make the inequality true. You can find the critical points by solving the inequality for equality.

**Example:** Graph the solution set of the inequality $x > 2$.

**Solution:** To find the critical point, we solve the inequality for equality:

```
x = 2
```

The critical point is $x = 2$. To graph the solution set, we draw a number line and mark the critical point with an open circle. Then, we shade the region to the right of the critical point, since $x$ must be greater than 2.

```
[Image of a number line with the critical point $x = 2$ marked with an open circle. The region to the right of the critical point is shaded.]
```

### Graphing the Solution Set of a Compound Inequality

To graph the solution set of a compound inequality, you need to first graph the solution set of each individual inequality. Then, you can combine the solution sets to find the solution set of the compound inequality.

**Example:** Graph the solution set of the compound inequality $x < 3$ and $x > -1$.

**Solution:** First, we graph the solution set of each individual inequality.

- To graph the solution set of $x < 3$, we find the critical point by solving the inequality for equality:

```
x = 3
```

The critical point is $x = 3$. We draw a number line and mark the critical point with an open circle. Then, we shade the region to the left of the critical point, since $x$ must be less than 3.

```
[Image of a number line with the critical point $x = 3$ marked with an open circle. The region to the left of the critical point is shaded.]
```

- To graph the solution set of $x > -1$, we find the critical point by solving the inequality for equality:

```
x = -1
```

The critical point is $x = -1$. We draw a number line and mark the critical point with an open circle. Then, we shade the region to the right of the critical point, since $x$ must be greater than -1.

```
[Image of a number line with the critical point $x = -1$ marked with an open circle. The region to the right of the critical point is shaded.]
```

To graph the solution set of the compound inequality $x < 3$ and $x > -1$, we combine the solution sets of the individual inequalities. The solution set of the compound inequality is the region that is shaded in both graphs.

```
[Image of a number line with the critical points $x = 3$ and $x = -1$ marked with open circles. The region between the critical points is shaded.]
```

## Conclusion

Graphing the solution set of a compound inequality is a two-step process. First, you need to graph the solution set of each individual inequality. Then, you can combine the solution sets to find the solution set of the compound inequality.

## FAQs

**What is a compound inequality?**

A compound inequality is an inequality that contains two or more inequalities joined by the word “and” or “or.”

**How do I find the critical points of an inequality?**

To find the critical points of an inequality, you need to solve the inequality for equality.

**How do I graph the solution set of an inequality?**

To graph the solution set of an inequality, you need to draw a number line and mark the critical points. Then, you need to shade the region that satisfies the inequality.

**How do I graph the solution set of a compound inequality?**

To graph the solution set of a compound inequality, you need to first graph the solution set of each individual inequality. Then, you need to combine the solution sets to find the solution set of the compound inequality.

**What is the difference between the solution set of an inequality and the graph of an inequality?**

The solution set of an inequality is the set of all values of the variable that make the inequality true. The graph of an inequality is the visual representation of the solution set.

.

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