a. Let's start by designating variables to represent the monthly mileage for Ana's car and motorcycle. Let [tex]\( x \)[/tex] be the number of miles Ana drives her car in a month, and let [tex]\( y \)[/tex] be the number of miles she drives her motorcycle in the same period. The condition that the combined mileage of both vehicles should not exceed 500 miles per month can be modeled by the inequality:
[tex]\[ x + y \leq 500 \][/tex]
b. To graph all the pairs [tex]\((x, y)\)[/tex] that satisfy the inequality [tex]\(x + y \leq 500\)[/tex], we will:
1. Draw a coordinate system with the x-axis representing car miles [tex]\(x\)[/tex] and the y-axis representing motorcycle miles [tex]\(y\)[/tex].
2. Identify the boundary of the inequality, which is when [tex]\( x + y = 500 \)[/tex]. This is a straight line with a slope of [tex]\(-1\)[/tex], passing through the points (0, 500) and (500, 0) where either the car or the motorcycle has zero mileage, respectively.
3. Plot the line on the graph by connecting these two points.
4. Since the inequality is [tex]\( x + y \leq 500 \)[/tex], we are interested in the area that includes these points and all points below the line, as this area represents all possible combinations of car and motorcycle mileages that do not exceed 500 miles when combined.
c. The strategy I used to draw the graph was as follows:
1. Determine the boundary line of the inequality by identifying two clear intercepts where one of the variables is zero.
2. Plot this boundary line on a graph.
3. Shade or highlight the region that represents solutions to the inequality. Since it is [tex]\( x + y \leq 500 \)[/tex], the shaded area is below and including the line on the graph.
4. Label the axes and provide a title or description, as well as indicate the feasible region if necessary.
