prove |x + y| is greater than or equal to |x| - |y|
To prove that |x + y| is greater than or equal to |x| - |y| for any real numbers x and y, we can consider two cases:
Case 1: x + y ≥ 0:
In this case, we have |x + y| = x + y, |x| = x, and |y| = y. Therefore, the inequality |x + y| ≥ |x| - |y| can be written as x + y ≥ x - y. By simplifying this inequality, we get 2y ≥ 0. This is always true since y can be positive, negative, or zero. Therefore, the inequality holds for this case.
Case 2: x + y < 0:
In this case, we have |x + y| = -(x + y), |x| = x, and |y| = -y. Therefore, the inequality |x + y| ≥ |x| - |y| can be written as -(x + y) ≥ x - (-y), which simplifies to -2x ≥ 2y. Dividing both sides by -2, we get x ≤ -y. This is always true since x + y < 0 implies that x is less than -y. Therefore, the inequality also holds for this case.
Since the inequality holds for both cases, we can conclude that |x + y| is greater than or equal to |x| - |y| for any real numbers x and y.