\( \left. \begin{array} { l } { x ^ { 2 } + x - 20 \leq 0 } \\ { ( x + 5 ) } \\ { x + 5 = 0 \quad x - 4 = 0 } \\ { x = - 5 \quad x = 4 \Rightarrow P \cdot C } \end{array} \right. \)

The inequality \( x^2 + x - 20 \leq 0 \) can be factored into \( (x - 4)(x + 5) \leq 0 \). This tells us that the expression is less than or equal to zero between the roots \( x = -5 \) and \( x = 4 \). Thus, the solution set is \( -5 \leq x \leq 4 \). In graphical terms, you can visualize this by plotting the parabola, which opens upwards and crosses the x-axis at these two points, indicating the intervals where the function dips below the x-axis. In real-life applications, such inequalities are used in various fields such as economics, physics, and engineering to determine feasible ranges for variables. For example, if this inequality represented a profit function, the interval between \( x = -5 \) and \( x = 4 \) might indicate the range of production levels where a company does not incur a loss. Thus, understanding how to solve and interpret inequalities is vital for making informed decisions in practical scenarios.