Theory
In this lesson, we will use an important result known as the Triangle Inequality:
|x| + |y| ≥ |x + y|
This inequality shows that the sum of the absolute values of two numbers is always greater than or equal to the absolute value of their sum. In geometry, this means the sum of the lengths of two sides of a triangle is always at least as large as the length of the third side.
Problems
- Solve the equation |2026 − x| + |x − 2026| = 2025.
- Solve the inequality |x + 2025| < |x − 2026|.
- Prove that if a + b + c + d > 0, a > c, and b > d, then |a + b| > |c + d|.
- Prove that there are no real numbers x, y, t such that all three inequalities hold at the same time: |x| < |y − t|, |y| < |t − x|, |t| < |x − y|.
- On the coordinate plane, plot all points whose coordinates satisfy the equation y² − |y| = x² − |x|.
- Sophie says that for any three points A, B, and C in the plane, the inequality AB ≥ |AC − BC| holds. Prove that she is correct.
- Emily claims that in any triangle, the length of any side does not exceed the semiperimeter. Prove that she is right.
- Two rivers (straight lines) — one with fresh water and one with salt water — intersect at an acute angle. Inside this acute angle stands Tom, who wants to collect both fresh and salt water and return to his starting point. How should he do this while travelling the minimum possible distance