graph the system of inequalities, and classify the figure created bu the solution region.
y¡Ü4x+4
y¡Ü-0.25x+4
y¡Ý4x-1
y¡Ý-0.25x-1
a)the shaded region is a plane minus a rectangle
b)there is no region common to all four inequalities
c)the shaded region is a rectangle
d)the shaded region is a rectangle
To graph the system of inequalities, we can start by graphing each individual inequality and then identify the overlapping regions.
1) The inequality y ≤ 4x + 4 can be graphed as a solid line with a shaded region below it.
- Plot the y-intercept at (0, 4), and use the slope of 4 to find additional points.
- Draw a solid line through these points, and shade the area below the line.
2) The inequality y ≤ -0.25x + 4 can be graphed in a similar way.
- Plot the y-intercept at (0, 4), and use the slope of -0.25 to find additional points.
- Draw a solid line through these points, and shade the area below the line.
3) The inequality y ≥ 4x - 1 can be graphed as a solid line with a shaded region above it.
- Plot the y-intercept at (0, -1), and use the slope of 4 to find additional points.
- Draw a solid line through these points, and shade the area above the line.
4) The inequality y ≥ -0.25x - 1 can be graphed in a similar way.
- Plot the y-intercept at (0, -1), and use the slope of -0.25 to find additional points.
- Draw a solid line through these points, and shade the area above the line.
Now, look for the overlapping shaded regions.
- The overlapping area between the first two inequalities forms a triangular region below both lines.
- The overlapping area between the last two inequalities forms a triangular region above both lines.
- However, there is no common region that satisfies all four inequalities.
Therefore, the correct answer is b) there is no region common to all four inequalities.