Asystem of inequalities is given. \( \left\{\begin{array}{l}y \leq x+4 \\ y>3 x-1\end{array}\right. \) Which graph shows the solution set for the system of inequ

To solve the system of inequalities, we need to understand the region defined by each inequality. The line \(y = x + 4\) represents a line with a slope of 1, and since the inequality is \(y \leq x + 4\), the region below this line is included. On the other hand, the inequality \(y > 3x - 1\) corresponds to a line with a steeper slope of 3, and the solution set consists of the area above this line. To find the solution set, graph both lines on the same plane. The solution to the system is where the shaded area of \(y \leq x + 4\) overlaps with the shaded area of \(y > 3x - 1\). Look for the section where the lower (or equal) boundary of the first inequality meets the upper boundary of the second, offering a common ground for potential solution points. When analyzing the graphs, remember that the line for \(y = x + 4\) will be solid (inclusive of points on the line), while the line for \(y = 3x - 1\) will be dashed because it’s greater than, not inclusive. This juxtaposition of shaded areas will reveal the valid solutions for the given inequalities! To further explore similar concepts and gain deeper insights into inequalities, you might enjoy reading about linear programming. It's a powerful mathematical method used for optimizing outcomes in various fields like economics and engineering. For a more thorough foundation, consider diving into resources like the book "Elementary Graphing" by Robert E. Marcy. It provides clear explanations and examples of how to graph inequalities and interpret the solution sets they define!