1. **Problem Statement:** We want to understand the concept of inequalities and how they relate to graphs, especially in the context of a triangle defined by three inequalities. 2. **What is an Inequality?** An inequality compares two expressions and shows that one is greater than, less than, greater than or equal to, or less than or equal to the other. For example, $y > \frac{x}{4}$ means $y$ is greater than $\frac{x}{4}$. 3. **Graphing Inequalities:** When we graph an inequality like $y > \frac{x}{4}$, we first graph the line $y = \frac{x}{4}$. This line divides the plane into two regions. The inequality $y > \frac{x}{4}$ means we shade the region above this line. 4. **Triangle Defined by Inequalities:** The triangle ABC is formed by the intersection of three inequalities: - $y > \frac{x}{4}$ (region above the line $y = \frac{x}{4}$) - $x \geq 2$ (region to the right of the vertical line $x = 2$) - $x + y \geq 12.5$ (region above the line $x + y = 12.5$) 5. **Why Inequalities Form a Triangle:** Each inequality represents a half-plane. The triangle is the common area where all three half-planes overlap. 6. **Summary:** To solve problems like this, graph each boundary line, determine which side satisfies the inequality, and find the intersection of all these regions. This intersection is the solution set, often a polygon like a triangle. **Final answer:** The three inequalities defining triangle ABC are: $$y > \frac{x}{4}, \quad x \geq 2, \quad x + y \geq 12.5$$