**Which Number Line Shows the Solution to the Inequality?**

Have you ever grappled with the challenge of determining which number line accurately represents the solution to an inequality? The seemingly straightforward task of identifying the correct number line can often become perplexing, leaving you feeling frustrated and lost. But fear not, for help is at hand!

**Pain Points of Which Number Line Shows the Solution to the Inequality**

The struggle to identify the correct number line stems from several common challenges. One major obstacle is the lack of understanding of how inequalities are depicted on a number line. Additionally, the absence of clear guidelines or conventions for representing inequalities can lead to confusion and conflicting interpretations.

**The Solution: Determining the Correct Number Line**

The key to resolving this dilemma lies in understanding the relationship between inequalities and number lines. Inequalities, which express relationships such as “greater than” or “less than,” are typically represented on a number line using open circles (indicating exclusion) or closed circles (indicating inclusion). Open circles on the number line indicate that the endpoint is not included in the solution, while closed circles indicate that the endpoint is included.

By paying close attention to the circles used to represent the inequality, you can accurately determine which number line shows the correct solution. If the inequality uses open circles, the solution will not include the endpoint; if it uses closed circles, the solution will include the endpoint.

**Summary**

To summarize, the correct number line for an inequality is determined by the type of circles used to represent the endpoint. Open circles indicate that the endpoint is not included in the solution, while closed circles indicate that the endpoint is included. By understanding this relationship, you can confidently identify the number line that accurately represents the solution to an inequality.

## solutions”>Number Lines and Inequality Solutions

**Introduction**

Inequalities are mathematical statements that express an unbalanced relationship between two expressions. One way to visualize the solution to an inequality is through a number line. A number line is a straight line marked with tick marks or numbers at regular intervals, representing the real numbers.

## Representing Inequalities on Number Lines

To represent an inequality on a number line, follow these steps:

- Draw a number line.
- Identify the inequality symbol (<, >, ≤, ≥).
- Plot the solution point on the number line.
- Shade the region of the number line that satisfies the inequality.

**Different Inequality Symbols and Shading**

| Inequality Symbol | Shading |

|—|—|

| < | Left of the solution point |
| > | Right of the solution point |

| ≤ | Includes the solution point |

| ≥ | Includes the solution point |

**Example 1: Number Line for x < 5**

In this example, x < 5. The solution point is 5. The region to the left of 5 is shaded because it satisfies the inequality.

## Solutions to Inequalities on Number Lines**

The solution to an inequality on a number line is the set of numbers that make the inequality true. The shaded region on the number line represents the solution set.

**Positive Inequalities (x > a)**

For an inequality like x > a, the solution set is all numbers greater than a. The number line is shaded to the right of the solution point a.

**Negative Inequalities (x < a)**

For an inequality like x < a, the solution set is all numbers less than a. The number line is shaded to the left of the solution point a.

**Non-Strict Inequalities (x ≥ a, x ≤ a)**

For non-strict inequalities, the solution set includes the solution point a. The number line is shaded both to the left and right of the solution point a, depending on the inequality symbol.

## Special Cases**

**Compound Inequalities**

Compound inequalities involve multiple inequalities combined using and/or. The solution set is the intersection or union of the individual solution sets, respectively.

**Absolute Value Inequalities**

Absolute value inequalities involve expressions with absolute values. The solution set is found by breaking the inequality into two cases, one for the positive and one for the negative value of the expression inside the absolute value.

**Conclusion**

Number lines are a useful tool for visualizing and solving inequalities. By understanding how to represent inequalities on a number line and interpret their solutions, students can effectively use number lines to determine the solution sets of various inequalities.

## FAQs

**Q1: What is the difference between a closed and an open circle on a number line?**

A1: A closed circle indicates that the solution point is included in the solution set, while an open circle indicates that the solution point is not included in the solution set.

**Q2: How do I represent an inequality that includes both positive and negative numbers?**

A2: Use the inequality symbol ≥ or ≤, which shades both the left and right sides of the solution point on the number line.

**Q3: What is the solution set of an inequality like x ≥ 5 but x < 10?**

A3: The solution set is all numbers between 5 and 10, including 5. It is represented on a number line as a shaded region from 5 to 10.

**Q4: Can number lines be used to solve inequalities with absolute values?**

A4: Yes, but the inequality must be broken into two cases, one for the positive value of the expression inside the absolute value and one for the negative value.

**Q5: What is the solution set of an inequality like |x – 3| > 2?**

A5: The solution set is all numbers less than 1 or greater than 5. It is represented on a number line as two shaded regions: one to the left of 1 and one to the right of 5.

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