For any real numbers x, y, and z, if z=x+|y| which of the following must always be true?
z-x = |y|
so z-x >= 0
z >= x
To determine which of the given statements must always be true, let's analyze the equation z = x + |y|.
The expression |y| represents the absolute value of y. This means that |y| will always be non-negative, regardless of the value of y. Therefore, z = x + |y| will always be greater than or equal to x.
Given this information, we can evaluate the statements:
1. z ≥ x: This statement is always true, as we determined that z will always be greater than or equal to x.
2. z ≤ x: This statement is not always true since we concluded that z is greater than or equal to x.
3. z > y: We cannot determine the relationship between z and y based on the given equation. Therefore, this statement is not guaranteed to be true.
4. z < y: Similarly, we cannot determine if z is always less than y from the given equation. Thus, this statement is not necessarily true.
Therefore, the only statement that must always be true is: z ≥ x.