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 and classify the figure created by the solution region, we can follow these steps:
1. Graph the equations of the boundary lines of each inequality:
- y ≤ 4x + 4
- y ≤ -0.25x + 4
- y ≥ 4x - 1
- y ≥ -0.25x - 1
2. To graph each equation, we'll start with the y-intercept by finding the value of y when x = 0, and then find the x-intercept by finding the value of x when y = 0.
For the first equation (y ≤ 4x + 4):
- y = 4x + 4
When x = 0: y = 4(0) + 4 = 4
When y = 0: 0 = 4x + 4 -> 4x = -4 -> x = -1
Plot the points (0,4) and (-1,0), and draw a line passing through these points.
For the second equation (y ≤ -0.25x + 4):
- y = -0.25x + 4
When x = 0: y = -0.25(0) + 4 = 4
When y = 0: 0 = -0.25x + 4 -> 0.25x = 4 -> x = 16
Plot the points (0,4) and (16,0), and draw a line passing through these points.
For the third equation (y ≥ 4x - 1):
- y = 4x - 1
When x = 0: y = 4(0) - 1 = -1
When y = 0: 0 = 4x - 1 -> 4x = 1 -> x = 0.25
Plot the points (0,-1) and (0.25,0), and draw a line passing through these points.
For the fourth equation (y ≥ -0.25x - 1):
- y = -0.25x - 1
When x = 0: y = -0.25(0) - 1 = -1
When y = 0: 0 = -0.25x - 1 -> 0.25x = -1 -> x = -4
Plot the points (0,-1) and (-4,0), and draw a line passing through these points.
3. Once all the lines are graphed, shade the region that satisfies all four inequalities. This shaded region represents the solution region.
Based on the description, it appears that option (c) is the correct answer: the shaded region is a rectangle.
To graph the system of inequalities, we first need to plot the boundary lines for each inequality.
1. Start with the first inequality: y ≤ 4x + 4.
- To plot the boundary line, first start with the equation y = 4x + 4.
- Choose a few values for x, calculate the corresponding y-values, and plot the points.
- Connect the points to graph the line and make sure to draw it as a dashed line since it is a "less than or equal to" inequality.
2. Move to the second inequality: y ≤ -0.25x + 4.
- Again, rewrite the equation as y = -0.25x + 4.
- Choose some x-values, calculate the corresponding y-values, and plot the points.
- Connect the points with a dashed line.
3. Next, consider the third inequality: y ≥ 4x - 1.
- Rewrite the equation as y = 4x - 1.
- Choose values for x, calculate the corresponding y-values, and plot the points.
- Connect the points with a solid line (since it is a "greater than or equal to" inequality).
4. Finally, let's focus on the last inequality: y ≥ -0.25x - 1.
- Rewrite the equation as y = -0.25x - 1.
- Choose values for x, calculate the corresponding y-values, and plot the points.
- Connect the points with a solid line.
Now that we have graphed the boundaries for all four inequalities, we can shade the region that satisfies all the conditions. The shaded region is the area where all the inequalities overlap.
To determine the shape of the shaded region, analyze the graph. In this case, the shaded region forms a rectangle.
Therefore, the correct answer is (c) the shaded region is a rectangle.