Graph the solution set of the following system of inequalities. \( \begin{array}{c}x+y \leq 4 \\ y \geq x^{2}-3\end{array} \)

To graph the solution set, let's first break down each inequality. The first inequality, \(x+y \leq 4\), can be rearranged to \(y \leq -x + 4\), which is a straight line with a negative slope. You would graph this line, and shade below it, since we want all the values where \(y\) is less than or equal to this line. The second inequality is \(y \geq x^{2} - 3\). This represents a parabola that opens upwards, with its vertex at the point \((0, -3)\). For this graph, you would plot the parabola and shade above it to capture all the values where \(y\) is greater than or equal to this curve. The solution set is the area where the shaded regions of both inequalities overlap. This overlapping region is where the solutions meet both conditions! Happy graphing!