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A graphing utility is a tool (like a graphing calculator or software) that helps visualize mathematical functions and their relationships. By plotting functions on a graph, you can observe where they intersect, which is especially useful for solving equations with multiple functions. In this exercise, you need to graph two functions, namely, \(y = \cos x\) and \(y = -\frac{x}{5}\). By doing so, you can visually spot where these graphs intersect, providing the solution to the problem. Graphing utilities can handle complicated functions and give you a deep understanding of function behavior.

Intersection points on graphs occur where two or more functions cross each other. Each intersection point represents a solution where the equations of the functions are equal. For our specific equation \(x+5\cos x=0\), we isolated the functions and plotted them as \(y = \cos x\) and \(y = -\frac{x}{5}\). The intersection points on this graph reveal the x-values that solve the equation. If you're using a graphing utility, look at the graph and identify the x-coordinates of these intersection points. Make sure to round these coordinates to two decimal places as the problem requires.

Trigonometric functions are fundamental in mathematics, describing the relationship between the angles and sides of triangles. In this particular problem, the trigonometric function is the cosine function. The equation given is \(x + 5\cos x = 0\), which you rearrange to \cos x = -\frac{x}{5}\. The cosine function (\textbackslash cos) oscillates between -1 and 1, producing a wave-like graph. Understanding these properties helps you anticipate how the graph will look and how it will interact with the linear function \(y = -\frac{x}{5}\). When solving trigonometric equations, appreciate how these functions behave over their period and amplitude to reliably find solutions by graphing.