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 Morgan's dog may be from the house?
A. 496 meters and 500 meters
B. 500 meters and 508 meters
C. 492 meters and 508 meters
Answer (1)
The problem gives the equation ∣ x − 500∣ = 8 .
Split the absolute value equation into two cases: x − 500 = 8 and x − 500 = − 8 .
Solve each case for x : x = 508 and x = 492 .
The minimum and maximum distances are 492 meters and 508 meters .
Explanation
Understanding the Problem We are given the equation ∣ x − 500∣ = 8 , which represents the distance of Morgan's dog from her house. We need to find the minimum and maximum possible values of x .
Splitting into Cases To solve the absolute value equation ∣ x − 500∣ = 8 , we consider two cases:
Case 1: x − 500 = 8 Case 2: x − 500 = − 8
Solving for x Solving for x in each case:
Case 1: x − 500 = 8 . Adding 500 to both sides gives x = 500 + 8 = 508 .
Case 2: x − 500 = − 8 . Adding 500 to both sides gives x = 500 − 8 = 492 .
Finding the Minimum and Maximum Distances Therefore, the minimum distance is 492 meters and the maximum distance is 508 meters.
Examples
Imagine you are tracking the location of a delivery drone. The drone is programmed to stay within a certain radius of a central hub. This problem helps determine the farthest and closest the drone can be from the hub, ensuring it stays within operational limits. Understanding absolute value equations helps define these boundaries, crucial for logistics and safety.
