Question

a contractor is using a magnetic stud finder to locate a nail in a stud behind a wall. when the stud finder is directly over the nail, its magnet is 1 cm away from the wall. the nail is 5 cm behind the wall. which 2 of the following expressions represents the distance between the nail and the magnet? choose 2 answers: a |-5 - 1| b |1 + 5| c |-1 + 5|

Explanation:

Step1: Model the situation

We can think of the wall as a reference (e.g., the origin on a number line). Let the wall be at position \(0\). The magnet is \(1\) cm away from the wall, so if we consider the direction towards the nail as positive, the magnet's position is \(1\) cm (or if we take the wall as a boundary, maybe the magnet is at \(1\) and the nail is at \(- 5\) if we consider the wall as \(0\) and behind the wall as negative, or vice - versa). The distance between two points \(x\) and \(y\) on a number line is given by \(|x - y|\).

Suppose the wall is at \(0\), the magnet is at \(x = 1\) (since it's \(1\) cm away from the wall towards the front, let's say), and the nail is at \(y=- 5\) (since it's \(5\) cm behind the wall). Then the distance \(d=|1-(-5)|=|1 + 5|\) or \(d = |-5 - 1|=|-(5 + 1)|=|5 + 1|\) (because \(|a|=|-a|\)).

Step2: Analyze each option

• Option A: \(|-5 - 1|=|-6| = 6\). The distance between the nail and the magnet should be the distance between two points. If we consider the magnet at \(1\) and the nail at \(-5\), \(|-5-1|=|-(5 + 1)| = 6\), which is the distance.
• Option B: \(|1 + 5|=|6| = 6\). Since the distance between \(1\) and \(-5\) is \(|1-(-5)|=|1 + 5| = 6\), this is also the distance.
• Option C: \(|-1+5|=|4| = 4\), which is not equal to the correct distance (which should be \(6\)).

Answer:

A. \(|-5 - 1|\), B. \(|1 + 5|\)