These statements describe a solution region for a system of linear inequalities:

The intersections of its three boundaries are at (0, 4), (6, 1), and (0, -2).
The boundary that is farthest left is dashed and vertical, and it intercepts the x-axis at (0,
0).
The boundary at the top is solid, with a y-intercept of 4 and a slope of -1/2.
The boundary at the bottom is solid, with a slope of 1/2 and an x-intercept of 4.
The solution region is found in two quadrants.
Use these statements to identify the system of linear inequalities, and then graph the system.

top

y = -.5 x + 4

bottom
y = .5 x + b and goes through (4, 0)
0 = 2 + b so b = -2
y = .5 x - 2

left is x = 0 vertical (the y axis)

so a triangle shaded inside in quadrants 1 and 4
x > 0
y </= -0.5 x +4
y >/= +0.5 x - 2

To identify the system of linear inequalities, let's analyze the given statements one by one and determine the equations for each boundary.

1. The boundary that is farthest left is dashed and vertical, and it intercepts the x-axis at (0, 0).
This implies that the left boundary is a vertical line parallel to the y-axis, passing through (0, 0). Since it is dashed, it indicates that the inequality is strict. Thus, the equation for the left boundary is x < 0.

2. The boundary at the top is solid, with a y-intercept of 4 and a slope of -1/2.
The top boundary is a solid line with a negative slope of -1/2 and a y-intercept of 4. The equation for this line can be found using the point-slope form: y = mx + b, where m is the slope and b is the y-intercept. Therefore, the equation for the top boundary is y = -1/2x + 4.

3. The boundary at the bottom is solid, with a slope of 1/2 and an x-intercept of 4.
The bottom boundary is a solid line with a positive slope of 1/2 and an x-intercept of 4. Again, we can use the point-slope form to find the equation for this line: y = mx + b. Plugging in the slope and the x-intercept, the equation for the bottom boundary is y = 1/2x - 2.

Now, let's graph these three equations on a coordinate plane to visualize the solution region.

First, draw the vertical dashed line x = 0. This will be the left boundary.

Next, plot the point (0, 4) and use the negative slope of -1/2 to draw the solid line: y = -1/2x + 4. Extend this line until it intersects the dashed vertical line.

Lastly, plot the point (4, 0) and use the positive slope of 1/2 to draw the solid line: y = 1/2x - 2. Extend this line until it intersects the dashed vertical line.

The solution region is the triangular region enclosed by the three boundaries. Shade this region.

The graph should have the left boundary as a dashed vertical line that intercepts the x-axis at (0,0), the top boundary as a solid line with a negative slope passing through (0,4), and the bottom boundary as a solid line with a positive slope intercepting the x-axis at (4,0). The solution region is in the two quadrants enclosed by these lines.