Short Answer 
The answer explains the concept of inequalities in mathematics and applies it to analyze a football tournament scenario for team Olympus, which plays 19 games. Using the scoring system, the team earns points and the inequalities indicate that Olympus must have at least 5 wins, leading to a breakdown of possible combinations of wins, ties, and losses.
Step 1: Understanding Inequality
An inequality is a mathematical statement that illustrates the relationship between two values. Specifically, it shows that one value is either greater than, less than, or equal to another. This can be expressed using symbols such as > (greater than), < (less than), ≥ (greater than or equal to), or ≤ (less than or equal to). Recognizing this concept is crucial in solving problems involving comparisons of numbers or algebraic expressions.
Step 2: Analyzing the Football Tournament Points
In the context of the football tournament, each team, including Olympus, plays exactly 19 games and earns points based on their performance. The scoring system is defined as follows: each win earns the team 3 points, each tie earns 1 point, and each loss earns 0 points. The total points for Olympus are given by the equation: 3w + t = 28, where w represents wins and t represents ties.
Step 3: Solving for Wins and Ties
To find the possible values for the number of wins and ties, we need to solve the inequalities based on the total games played. The essential inequalities are: T + W ≤ 19 and it’s also established that T, W ‚â• 0. By manipulating these equations, we find that the team must have at least 5 wins. Based on the results, the ties for Olympus can be defined in a list as follows:
- Wins: 5, Ties: 13, Losses: 1
- Wins: 6, Ties: 10, Losses: 3
- Wins: 7, Ties: 7, Losses: 5
- Wins: 8, Ties: 4, Losses: 7
- Wins: 9, Ties: 1, Losses: 9
This analysis reveals the performance of Olympus in the tournament.