what is the graph of the system ?y<-x-1 y>x+4
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To determine the graph of the system of inequalities, y < -x - 1 and y > x + 4, you can follow these steps:
Step 1: Begin by graphing the line y = -x - 1. This is a straight line with a slope of -1 and y-intercept of -1. Plot two points on the line, such as (-2, 1) and (0, -1), and draw the line connecting these points.
Step 2: Next, draw a dashed line above the line y = -x - 1 to represent the inequality y < -x - 1. This means that the area under this line is not included in the solution set.
Step 3: Now, graph the line y = x + 4. This is a straight line with a slope of 1 and a y-intercept of 4. Plot two points on the line, such as (-2, 2) and (0, 4), and draw the line connecting these points.
Step 4: Finally, draw a dashed line below the line y = x + 4 to represent the inequality y > x + 4. This means that the area above this line is not included in the solution set.
Step 5: The shaded region where the two dashed lines do not overlap represents the solution to the system of inequalities y < -x - 1 and y > x + 4. This shaded region is the area between the two lines.
Note: The lines that represent the inequalities are dashed to indicate that they are not included in the solution set. The shaded region represents the solution set of the system of inequalities.
To understand the graph of the system of inequalities y < -x - 1 and y > x + 4, we can follow these steps:
Step 1: Let's start by graphing the equation y = -x - 1. This is a straight line with a slope of -1 (negative slope) and a y-intercept (where the line crosses the y-axis) of -1. We can plot a few points on this line to help us visualize it. For example, when x = 0, y = -1, and when x = -1, y = 0. Connect these points to create the line.
Step 2: Next, let's graph the equation y = x + 4. This is another straight line, but with a positive slope of 1 and a y-intercept of 4. We can plot a few points on this line as well. For instance, when x = 0, y = 4, and when x = -4, y = 0. Connect these points to form the line.
Step 3: Now, let's analyze the inequality y < -x - 1. Since it is less than, not less than or equal to, the line y = -x - 1 is a dashed line. This means the boundary line is not included in the solution set.
Step 4: To determine which side of the line to shade, we can choose a point not on the line and check if it satisfies the inequality. One convenient point is the origin (0, 0). Substitute the coordinates x = 0 and y = 0 into the inequality y < -x - 1. We get 0 < -0 - 1, which simplifies to 0 < -1. Since this statement is false, the region below the line is not part of the solution.
Step 5: Now, let's consider the inequality y > x + 4. Like the previous equation, this inequality is not less than or equal to, so the line y = x + 4 is also a dashed line.
Step 6: Choose another point not on the line to check which side to shade. Let's use the origin again (0, 0). Substituting x = 0 and y = 0 into the inequality y > x + 4, we have 0 > 0 + 4, which simplifies to 0 > 4. Since this statement is false, the region above the line is not part of the solution.
Step 7: Finally, the shaded region that satisfies both inequalities is the region between the two dashed lines. This region represents all the points that satisfy both y < -x - 1 and y > x + 4.
To summarize, the graph of the system of inequalities y < -x - 1 and y > x + 4 consists of the shaded region between the dashed lines y = -x - 1 and y = x + 4.