Question

morgan is walking her dog on an 8-meter-long leash. she is currently 500 meters from her house, so the maximum and minimum distances that the dog may be from the house can be found using the equation |x - 500| = 8. what are the minimum and maximum distances that morgans dog may be from the house?
496 meters and 500 meters
500 meters and 508 meters
496 meters and 504 meters
492 meters and 508 meters

Explanation:

Step1: Solve the absolute value equation

The equation is \(|x - 500| = 8\). By the definition of absolute value, this means \(x - 500 = 8\) or \(x - 500 = -8\).

Step2: Find the maximum distance

For the maximum distance, we solve \(x - 500 = 8\). Adding 500 to both sides gives \(x = 500 + 8 = 508\).

Step3: Find the minimum distance

For the minimum distance, we solve \(x - 500 = -8\). Adding 500 to both sides gives \(x = 500 - 8 = 492\). Wait, no, wait. Wait, Morgan is 500 meters from the house, and the leash is 8 meters. So the dog can be 500 - 8 = 492? No, wait, no. Wait, the equation is \(|x - 500| = 8\), which means the distance \(x\) from the house differs from 500 by 8. So the minimum distance is \(500 - 8 = 492\)? No, that can't be. Wait, no, Morgan is at 500 meters from the house. The dog's distance from the house, \(x\), has \(|x - 500| = 8\). So \(x = 500 + 8 = 508\) (maximum, when the dog is away from the house from Morgan's position) and \(x = 500 - 8 = 492\)? Wait, but that would mean the dog is closer to the house than Morgan? But if Morgan is 500 meters from the house, and the leash is 8 meters, the dog can go towards the house (so closer, 500 - 8 = 492) or away from the house (500 + 8 = 508). Wait, but the options: let's check the options. The options are:

1. 496 meters and 500 meters – no.

1. 500 meters and 508 meters – no.

1. 496 meters and 504 meters – no.

1. 492 meters and 508 meters – yes, that's the fourth option. Wait, but wait, my initial calculation was wrong earlier. Wait, solving \(|x - 500| = 8\):

Case 1: \(x - 500 = 8\) → \(x = 508\) (maximum).

Case 2: \(x - 500 = -8\) → \(x = 492\) (minimum). So the minimum distance is 492, maximum is 508. So the fourth option is 492 meters and 508 meters. Wait, but let's re - evaluate. Wait, maybe I misread the problem. The problem says "Morgan is walking her dog on an 8 - meter - long leash. She is currently 500 meters from her house". So Morgan's position is 500 meters from the house. The dog's distance from the house, \(x\), must satisfy \(|x - 500| = 8\). So \(x = 500\pm8\). So \(x = 508\) (max) and \(x = 492\) (min). So the correct option is "492 meters and 508 meters". Wait, but let's check the options again. The options given are:

• 496 meters and 500 meters

• 500 meters and 508 meters

• 496 meters and 504 meters

• 492 meters and 508 meters

Ah, so the fourth option is 492 meters and 508 meters. Wait, but my earlier step 3 had a miscalculation before, but now corrected. So the maximum is 508, minimum is 492. So the correct option is the fourth one: 492 meters and 508 meters.

Wait, but let's do the equation again. \(|x - 500| = 8\)

\(x - 500 = 8\) → \(x = 508\)

\(x - 500 = -8\) → \(x = 492\)

Yes, so the minimum distance is 492 meters, maximum is 508 meters. So the answer is the fourth option: 492 meters and 508 meters.

Wait, but let's check the options again. The options are:

1. 496 meters and 500 meters – incorrect.

1. 500 meters and 508 meters – incorrect.

1. 496 meters and 504 meters – incorrect.

1. 492 meters and 508 meters – correct.

So the correct answer is the fourth option: 492 meters and 508 meters.

Wait, but in my initial steps, I made a mistake in the first calculation, but now corrected. So the steps are:

Step1: Interpret the absolute value equation

\(|x - 500| = 8\) means \(x - 500 = 8\) or \(x - 500 = -8\).

Step2: Solve for maximum distance (\(x - 500 = 8\))

\(x = 500 + 8 = 508\).

Step3: Solve for minimum distance (\(x - 500 = -8\))

\(x = 500 - 8 = 492\).

So the minimum distance is 492 meters and the maximum d…

Answer:

492 meters and 508 meters (the fourth option: 492 meters and 508 meters)