System Of Linear Inequalities

Question

y <= -15x + 3000

y <= 5x

In the xy plane, if a point with coordinates (a,b) lies in the solution set of the system of inequalities above, what is the maximum possible value for b?

Answer 1

The maximum value for b would occur at the point of intersection between the two inequalities. It's easy to recognize this by picturing in our heads what the system looks like. The first inequality would be a line starting at y = 3000 and decreasing quickly. The second inequality would be a line starting at y = 0 and rising slowly. The point of intersection would therefore contain the highest y-value that satisfies the system. To find the point of intersection, we just have to set the two inequalities equal to each other:

-15x + 3000 = 5x

3000 = 20x (added 15x to both sides)

x = 150 (divided 20 on both sides)

To find b, simply plug x into either of the two inequalities. It's easier to solve 5x versus -15x + 3000 so we will solve for 5x which would be 5(150) = 750 = b which is our answer.