Question

1. dan and kyle both go to the same gym. dan lives 35 miles from the gym. kyle lives 8 miles from dan.

part a. write an absolute value equation to represent the location of kyle’s house in relation to the gym.

part b. how far could kyle live from the gym?

Explanation:

Part A

Step1: Define Variables

Let \( x \) be the distance from Kyle's house to the gym, and \( 35 \) be Dan's distance to the gym. The distance between Kyle and Dan is \( 8 \), so \( |x - 35| = 8 \).

Step2: Verify the Equation

The absolute value represents the distance between two points, so the difference in their distances to the gym (or vice versa) should equal the distance between them. Thus, \( |x - 35| = 8 \) models this.

Step1: Solve the Absolute Value Equation

For \( |x - 35| = 8 \), we have two cases:

• Case 1: \( x - 35 = 8 \), so \( x = 35 + 8 = 43 \).
• Case 2: \( x - 35 = -8 \), so \( x = 35 - 8 = 27 \).

Step2: Interpret the Solutions

These solutions represent the two possible distances (Kyle could be 8 miles towards or away from the gym relative to Dan's position).

Answer:

\( |x - 35| = 8 \)

Part B