Question

1. (a.ei.2, 10 points)

the solutions to a system of linear inequalities and points f, g, h, j, k, l, m, and n are shown. all labeled points have integral coordinates.

• points f, j, and m each lie on a boundary line.
• point g lies on both boundaries.

circle each labeled point that represents a solution to this system of linear inequalities.
graph with points f, g, h, j, k, l, m, n on a coordinate plane with shaded regions and boundary lines

Explanation:

Step1: Recall solution region rules

The solution to a system of linear inequalities is the region (including boundaries if inequalities are non - strict) where all inequalities are satisfied. Points on the boundary lines (if the inequalities are $\leq$ or $\geq$) and inside the overlapping shaded region are solutions.

Step2: Analyze each point

• Point F: Lies on a boundary line and in the shaded region (since the shaded area includes the boundary it's on), so it's a solution.
• Point G: Lies on both boundaries (intersection of boundaries), so it's a solution (as intersection of boundaries is part of the solution set for non - strict inequalities).
• Point H: Inside the shaded region, so it's a solution.
• Point J: Lies on a boundary line and in the shaded region, so it's a solution.
• Point K: Outside the shaded region, so not a solution.
• Point L: Outside the shaded region, so not a solution.
• Point M: Lies on a boundary line and in the shaded region, so it's a solution.
• Point N: Inside the shaded region, so it's a solution.

Answer:

The points that are solutions are F, G, H, J, M, N. (To circle them, identify each point: F (on y - axis, y = 2 or so), G (intersection of two boundaries), H (left - upper shaded), J (on lower boundary, in shaded), M (on left - lower boundary, in shaded), N (upper - middle shaded))