The Minimum and Maximum Distances

Oct 13, 2025 by windailys.com 136 views

Introduction

In this article, we will delve into a mathematical problem involving a dog on a leash and its owner. Morgan is walking her dog on an 8-meter-long leash, and she is currently 500 meters from her house. We need to find the maximum and minimum distances that the dog may be from the house using the equation $|x-500|=8$ . This problem requires us to understand absolute value equations and how they can be used to solve real-world problems.

What is an Absolute Value Equation?

An absolute value equation is an equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction. In other words, it is the magnitude of the number. For example, the absolute value of -3 is 3, and the absolute value of 3 is also 3.

The Equation $|x-500|=8$

The equation $|x-500|=8$ represents the distance between the dog and the house. The absolute value of $x-500$ represents the distance between the dog and the house, and this distance is equal to 8 meters. We can rewrite this equation as two separate equations:

$x-500=8$ and $x-500=-8$

Solving the Equations

Let's solve the two equations separately.

Solving $x-500=8$

To solve this equation, we need to isolate the variable $x$ . We can do this by adding 500 to both sides of the equation:

$x-500+500=8+500$

This simplifies to:

$x=508$

So, the first solution is $x=508$ .

Solving $x-500=-8$

To solve this equation, we need to isolate the variable $x$ . We can do this by adding 500 to both sides of the equation:

$x-500+500=-8+500$

This simplifies to:

$x=492$

So, the second solution is $x=492$ .

The Minimum and Maximum Distances

The minimum distance that the dog may be from the house is 492 meters, and the maximum distance that the dog may be from the house is 508 meters.

Conclusion

In this article, we explored a mathematical problem involving a dog on a leash and its owner. We used absolute value equations to find the minimum and maximum distances that the dog may be from the house. We learned how to solve absolute value equations and how they can be used to solve real-world problems.

Real-World Applications

Absolute value equations have many real-world applications. For example, they can be used to model the distance between two objects, the cost of a product, or the temperature of a system. In this problem, we used absolute value equations to find the minimum and maximum distances that the dog may be from the house.

Tips and Tricks

When solving absolute value equations, it's essential to remember that the absolute value of a number is its distance from zero on the number line, without considering direction. This means that the absolute value of a negative number is always positive.

Common Mistakes

When solving absolute value equations, it's common to make mistakes such as:

Not isolating the variable
Not considering the direction of the number
Not using the correct absolute value equation

Practice Problems

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

Solve the equation $|x+2|=5$
Solve the equation $|x-3|=2$
Solve the equation $|x+1|=4$

Answer Key

$x=-7$ or $x=3$
$x=1$ or $x=5$
$x=-5$ or $x=5$

Conclusion

Introduction

In our previous article, we explored a mathematical problem involving a dog on a leash and its owner. We used absolute value equations to find the minimum and maximum distances that the dog may be from the house. In this article, we will provide a Q&A guide to help you understand absolute value equations and how to solve them.

Q: What is an absolute value equation?

A: An absolute value equation is an equation that involves the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line, without considering direction.

Q: How do I solve an absolute value equation?

A: To solve an absolute value equation, you need to isolate the variable and then consider the two possible cases:

Case 1: The expression inside the absolute value is positive.
Case 2: The expression inside the absolute value is negative.

Q: What is the difference between the two cases?

A: In Case 1, the absolute value equation becomes a simple linear equation. In Case 2, the absolute value equation becomes a linear equation with a negative coefficient.

Q: How do I find the solutions to an absolute value equation?

A: To find the solutions to an absolute value equation, you need to solve the two cases separately and then combine the solutions.

Q: What are some common mistakes to avoid when solving absolute value equations?

A: Some common mistakes to avoid when solving absolute value equations include:

Not isolating the variable
Not considering the direction of the number
Not using the correct absolute value equation

Q: Can you provide some examples of absolute value equations?

A: Here are some examples of absolute value equations:

$|x+2|=5$
$|x-3|=2$
$|x+1|=4$

Q: How do I solve the equation $|x+2|=5$ ?

A: To solve the equation $|x+2|=5$ , you need to isolate the variable and then consider the two possible cases:

Case 1: $x+2=5$
Case 2: $x+2=-5$

Solving these two cases separately, you get:

$x=3$
$x=-7$

Q: How do I solve the equation $|x-3|=2$ ?

A: To solve the equation $|x-3|=2$ , you need to isolate the variable and then consider the two possible cases:

Case 1: $x-3=2$
Case 2: $x-3=-2$

Solving these two cases separately, you get:

$x=5$
$x=1$

Q: How do I solve the equation $|x+1|=4$ ?

A: To solve the equation $|x+1|=4$ , you need to isolate the variable and then consider the two possible cases:

Case 1: $x+1=4$
Case 2: $x+1=-4$

Solving these two cases separately, you get:

$x=3$
$x=-5$

Conclusion

In this article, we provided a Q&A guide to help you understand absolute value equations and how to solve them. We covered common mistakes to avoid, examples of absolute value equations, and step-by-step solutions to some of these equations. We hope this guide has been helpful in your understanding of absolute value equations.

Practice Problems

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

Solve the equation $|x-2|=3$
Solve the equation $|x+4|=2$
Solve the equation $|x-1|=5$

Answer Key

$x=5$ or $x=-1$
$x=6$ or $x=-2$
$x=6$ or $x=-4$