Final answer:
The solution set to the system of inequalities is identified by testing each point against the inequalities. Only option b. (-2.5,0) satisfies all the given inequalities, making it the correct solution.
Step-by-step explanation:
To determine the solution set to the system of inequalities given by the functions f(x) ≥ 2x + 2, f(x) ≤ -4x + 3, and f(x) ≤ 6x + 5, we need to find the set of values for x that satisfy all three inequalities simultaneously.
Since f(x) is bounded by these linear inequalities, we can graph the lines y = 2x + 2, y = -4x + 3, and y = 6x + 5 to visualize the regions they form. However, the options given are points on the coordinate plane, so we need to determine which of the points lie within the intersection of the regions defined by the inequalities.
After substituting the x and y values from each option into the inequalities and checking for satisfaction of all conditions, we find the following:
- For option a. (0,0), f(x) does not satisfy the first inequality: 0 ≥ 2(0) + 2.
- For option b. (-2.5,0), f(x) satisfies all inequalities: 0 ≥ 2(-2.5) + 2, 0 ≤ -4(-2.5) + 3, and 0 ≤ 6(-2.5) + 5.
- For option c. (0,6.5), f(x) satisfies the first inequality but not the others.
- For option d. (0,2.5), f(x) satisfies the first inequality but not the others.
Therefore, the solution set to the system of inequalities is represented by option b. (-2.5,0).