Shade above y ≥ 4/(x-2) and below y ≤ -x+5. Their overlap forms the solution: a triangle bounded by x-axis, y = 4/(x-2), and y = -x+5.
The system of inequalities is represented by the following two equations:
(x−2)(y−4)≥0
x+y≤5
We can solve the system by first graphing each inequality individually. Then, we can identify the shaded region that satisfies both inequalities simultaneously.
Steps to solve:
1. Graph the first inequality:
Rewrite the first inequality as y≥ 4/(x−2) .
This is the equation of a horizontal line passing through the point (2,4).
Since the inequality is non-strict, the line should be a dashed line.
Shade the region above the line (y values greater than or equal to 4/(x - 2)).
2. Graph the second inequality:
Rewrite the second inequality as y≤−x+5.
This is the equation of a sloped line passing through the points (0,5) and (5,0).
Shade the region below the line (y values less than or equal to -x + 5).
3. Identify the solution:
The solution to the system of inequalities is the region that overlaps both shaded areas.
In this case, the solution is the triangular region bounded by the x-axis, the line y= 4/(x-2) , and the line y=−x+5.
Answer:
The solution to the system of inequalities is the triangular region bounded by the x-axis, the line y=
x−2 and the line y=−x+5.
Additional notes:
The points (2,4), (5,−1), and (4,3) are all located within the solution region.
The point (0,0) is not part of the solution region because it does not satisfy both inequalities.