Representing the Walking Distance

Oct 13, 2025 by magicalceylontours.com 78 views

Jamaal's house is located on the 57th block of the town, and he is allowed to walk no farther than three blocks in either direction from his house. This means that Jamaal can walk up to three blocks north or south of his house. In this article, we will explore the absolute value equation that can be used to determine the farthest block to which Jamaal is allowed to walk.

Defining the Problem

To find the farthest block to which Jamaal is allowed to walk, we need to consider the possible locations of his walking distance. Since Jamaal can walk up to three blocks in either direction from his house, we can represent the possible locations as a range of blocks. The starting point is Jamaal's house, which is located on the 57th block. The farthest block to which Jamaal is allowed to walk will be either three blocks north or three blocks south of his house.

Representing the Walking Distance

Let's represent the farthest block to which Jamaal is allowed to walk as x. Since Jamaal can walk up to three blocks in either direction from his house, we can write an absolute value equation to represent this situation. The absolute value equation will be in the form |x - 57| ≤ 3, where x is the farthest block to which Jamaal is allowed to walk.

Solving the Absolute Value Equation

To solve the absolute value equation |x - 57| ≤ 3, we need to consider two cases:

Case 1: x - 57 ≤ 3
Case 2: x - 57 ≥ -3

Case 1: x - 57 ≤ 3

For Case 1, we can add 57 to both sides of the inequality to get x ≤ 60.

Case 2: x - 57 ≥ -3

For Case 2, we can add 57 to both sides of the inequality to get x ≥ 54.

Combining the Cases

Since the absolute value equation |x - 57| ≤ 3 represents the possible locations of Jamaal's walking distance, we need to combine the two cases to get the final solution. The final solution is x ≥ 54 and x ≤ 60.

Conclusion

In conclusion, the absolute value equation that can be used to determine the farthest block to which Jamaal is allowed to walk is |x - 57| ≤ 3. By solving this equation, we found that the farthest block to which Jamaal is allowed to walk is between 54 and 60 blocks.

Understanding Absolute Value Equations

Absolute value equations are used to represent the distance between two values. In this case, the absolute value equation |x - 57| ≤ 3 represents the distance between Jamaal's house (57th block) and the farthest block to which he is allowed to walk. The absolute value equation is used to find the range of possible values for x, which represents the farthest block to which Jamaal is allowed to walk.

Real-World Applications

Absolute value equations have many real-world applications, including:

Distance and Time: Absolute value equations can be used to represent the distance between two points and the time it takes to travel between them.
Finance: Absolute value equations can be used to represent the value of an investment and the risk associated with it.
Science: Absolute value equations can be used to represent the distance between two objects and the time it takes for them to collide.

Tips and Tricks

Here are some tips and tricks for solving absolute value equations:

Use the definition of absolute value: The absolute value of a number is its distance from zero on the number line.
Use the properties of absolute value: The absolute value of a number is always non-negative.
Use the definition of absolute value to rewrite the equation: Rewrite the equation in terms of the absolute value of the expression.

Practice Problems

Here are some practice problems to help you understand absolute value equations:

Problem 1: Solve the absolute value equation |x + 2| ≤ 5.
Problem 2: Solve the absolute value equation |x - 3| ≥ 2.
Problem 3: Solve the absolute value equation |x + 1| ≤ 4.

Conclusion

In our previous article, we explored the absolute value equation that can be used to determine the farthest block to which Jamaal is allowed to walk. In this article, we will answer some frequently asked questions about Jamaal's walking distance limitations.

Q: What is the absolute value equation that can be used to determine the farthest block to which Jamaal is allowed to walk?

A: The absolute value equation that can be used to determine the farthest block to which Jamaal is allowed to walk is |x - 57| ≤ 3.

Q: How do I solve the absolute value equation |x - 57| ≤ 3?

A: To solve the absolute value equation |x - 57| ≤ 3, you need to consider two cases:

Case 1: x - 57 ≤ 3
Case 2: x - 57 ≥ -3

By solving these two cases, you will get the final solution, which is x ≥ 54 and x ≤ 60.

Q: What does the absolute value equation |x - 57| ≤ 3 represent?

A: The absolute value equation |x - 57| ≤ 3 represents the distance between Jamaal's house (57th block) and the farthest block to which he is allowed to walk.

Q: Can I use the absolute value equation |x - 57| ≤ 3 to find the farthest block to which Jamaal is allowed to walk if his house is located on a different block?

A: Yes, you can use the absolute value equation |x - 57| ≤ 3 to find the farthest block to which Jamaal is allowed to walk if his house is located on a different block. However, you need to replace 57 with the actual block number where Jamaal's house is located.

Q: How do I use the absolute value equation |x - 57| ≤ 3 to find the farthest block to which Jamaal is allowed to walk if he is allowed to walk more than three blocks in either direction from his house?

A: If Jamaal is allowed to walk more than three blocks in either direction from his house, you need to replace 3 with the actual number of blocks that Jamaal is allowed to walk.

Q: Can I use the absolute value equation |x - 57| ≤ 3 to find the farthest block to which Jamaal is allowed to walk if he is allowed to walk in a different direction (e.g., east or west)?

A: Yes, you can use the absolute value equation |x - 57| ≤ 3 to find the farthest block to which Jamaal is allowed to walk if he is allowed to walk in a different direction (e.g., east or west). However, you need to replace 57 with the actual block number where Jamaal's house is located and adjust the equation accordingly.

Q: What are some real-world applications of absolute value equations?

A: Absolute value equations have many real-world applications, including:

Distance and Time: Absolute value equations can be used to represent the distance between two points and the time it takes to travel between them.
Finance: Absolute value equations can be used to represent the value of an investment and the risk associated with it.
Science: Absolute value equations can be used to represent the distance between two objects and the time it takes for them to collide.

Q: How do I practice solving absolute value equations?

A: You can practice solving absolute value equations by working on the following problems:

Problem 1: Solve the absolute value equation |x + 2| ≤ 5.
Problem 2: Solve the absolute value equation |x - 3| ≥ 2.
Problem 3: Solve the absolute value equation |x + 1| ≤ 4.

Conclusion

In conclusion, absolute value equations are used to represent the distance between two values. In this article, we answered some frequently asked questions about Jamaal's walking distance limitations and provided some practice problems to help you understand absolute value equations.