1. **Stating the problem:** We are given a geometry problem involving inequalities (inequations). We need to analyze the geometric conditions and solve the inequalities accordingly. 2. **Understanding the problem:** Typically, geometry problems with inequalities involve finding regions that satisfy certain conditions, such as distances, angles, or coordinates that meet inequality constraints. 3. **General approach:** - Identify the geometric figures involved (lines, circles, polygons). - Translate the geometric conditions into algebraic inequalities. - Use algebraic methods to solve these inequalities. 4. **Example:** Suppose we have a point $P(x,y)$ and a line $L: ax + by + c = 0$. The inequality $ax + by + c \leq 0$ represents all points on one side of the line. 5. **Solving inequalities:** - For linear inequalities, isolate variables step-by-step. - For quadratic or higher-degree inequalities, factorize or use sign analysis. 6. **Intermediate work example:** If we have $2x - 3y + 6 > 0$, we can write: $$ 2x - 3y + 6 > 0 $$ Isolate $y$: $$ -3y > -2x - 6 $$ Apply cancellation with sign change (dividing by $-3$ reverses inequality): $$ \cancel{-3}y < \cancel{-3}\frac{-2x - 6}{-3} $$ Simplify: $$ y < \frac{2}{3}x + 2 $$ This inequality describes the region below the line $y = \frac{2}{3}x + 2$. 7. **Summary:** - Translate geometric conditions to inequalities. - Solve inequalities stepwise, showing cancellations. - Interpret solutions as geometric regions. **Final answer:** The solution to the geometric inequalities is the set of points $(x,y)$ satisfying the derived inequalities, such as $y < \frac{2}{3}x + 2$ in the example.