Answer:
On a coordinate plane, the graph that shows the solution to the system of linear inequalities y < 2x – 5 and y > –3x + 1 is:
The first solid line has a negative slope and goes through (0, 1) and (1, negative 2). Everything to the right of the line is shaded. The second dashed line has a positive slope and goes through (0, negative 5) and (2, negative 1). Everything to the left of the line is shaded.
This shading represents the area where both inequalities are true. The solution to the system of linear inequalities is therefore the overlapping region to the left of the second line and to the right of the first line.
The solution to a system of linear inequalities is the region of the coordinate plane where all the inequalities are simultaneously satisfied. In this case, the two inequalities are y < 2x – 5 and y > –3x + 1.
The first inequality, y < 2x – 5, represents a line with a slope of 2 that passes through the points (0, -5/2) and (5/2, 0). The inequality indicates that all the points below this line satisfy the inequality. Therefore, the shaded region for this inequality is below the line.
The second inequality, y > –3x + 1, represents a line with a slope of -3 that passes through the points (0, 1) and (1/3, 0). The inequality indicates that all the points above this line satisfy the inequality. Therefore, the shaded region for this inequality is above the line.
The solution to the system of linear inequalities is the region where both shaded regions overlap, which is the region to the right of the first line and to the left of the second line. This region can be described as the set of all points (x, y) that satisfy the inequalities:
y < 2x – 5 and y > –3x + 1
or equivalently:
-3x + 1 < y < 2x – 5
The solution to this system of linear inequalities is a bounded region in the coordinate plane, and it can be represented graphically as the shaded trapezoidal region that is formed by the overlapping shaded regions of the two inequalities.