Final answer:
To solve the inequalities x + y > 0 and x + y + 5 < 0 graphically, graph the lines y = -x and y = -x - 5, and shade the appropriate regions.
Step-by-step explanation:
To solve the system of inequalities x + y > 0 and x + y + 5 < 0 graphically, follow these steps:
- First, graph the line x + y = 0. This line passes through the origin (0,0) and has a slope of -1 since it can be rewritten as y = -x. Label the axes with f(x) for the y-axis and x for the x-axis. For a useful scale, you might choose a range for x and y from -10 to 10, as the maximum value given for f(x) = 10,0 and the inequality involves small coefficients.
- Because the inequality is x + y > 0, shade the region above the line, which is where the inequality holds true.
- Next, graph the line x + y + 5 = 0, which can be rewritten as y = -x - 5. This line is parallel to the first and is shifted down by 5 units.
- For the inequality x + y + 5 < 0, shade the region below this second line.
- The solution to the system is the region where the shaded areas from both inequalities overlap. However, for this system of inequalities, you will find that there is no overlap, meaning there is no solution to this system.
Graphical methods can be useful for visualizing solutions, but analytical methods are usually more precise. This is particularly true when graphing is done by hand since it's harder to determine the exact point of intersection and the precise area of overlap.