Question

For any two real numbers, x and y, such that y ≠ 0, |x/y| = |x|/|y|. One of the cases in the proof of the theorem says: Since |x| = x and |y| = -y, |x|/|y| = x/(-y) = -x/y = |x/y|. Select the case that corresponds to this argument.

Answer 1

The case that corresponds to the provided argument is when |x| = x and |y| = -y.

Step-by-step explanation:

In the given argument, the case |x| = x and |y| = -y is selected. According to the properties of absolute values, when x is a positive real number, |x| is equal to x. Similarly, when y is a negative real number, |y| is equal to -y. The provided argument utilizes these properties to demonstrate that |x/y| is indeed equal to |x|/|y|.

The proof begins by stating that |x| = x and |y| = -y. This aligns with the case where x is positive, and y is negative. Substituting these values into the expression |x|/|y|, we get x/(-y). Simplifying further, we obtain -x/y. The key step in the argument is recognizing that this expression is equal to |x/y|. Therefore, the case |x| = x and |y| = -y is chosen to establish the equality |x/y| = |x|/|y|.

Understanding the properties of absolute values is crucial in proving the given theorem. The argument demonstrates the application of these properties in a specific case to establish the equality between the two expressions. This approach showcases the elegance and versatility of mathematical reasoning, allowing for a concise and rigorous proof of the stated theorem.