Which graph represents this system of inequalities? y ≤ 2x + 3 3x + y ≤ –2 (1 point) Responses The first line goes from left-parenthesis negative 3 comma negative 3 right-parenthesis to left-parenthesis 0 comma 3 right-parenthesis. The second line goes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma negative 2 right-parenthesis. The coordinate plane is shaded below the first line and to the left of the second line. Image with alt text: A coordinate plane is shown with two solid boundary lines graphed with their intersection in the second quadrant. Overlapping shaded region includes the points (negative 1, 0), (negative 2, negative 3), and (0, negative 5). The first line goes from left-parenthesis negative 3 comma negative 3 right-parenthesis to left-parenthesis 0 comma 3 right-parenthesis. The second line goes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma negative 2 right-parenthesis. The coordinate plane is shaded below the first line and to the left of the second line. The first line goes from left-parenthesis negative 3 comma negative 3 right-parenthesis to left-parenthesis 0 comma 3 right-parenthesis. The second line goes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma negative 2 right-parenthesis. The coordinate plane is shaded above the first line and to the left of the second line. Image with alt text: A coordinate plane is shown with two solid boundary lines graphed with their intersection in the second quadrant. Overlapping shaded region includes the points (negative 4, 4), (negative 3, 3), and (negative 2, 0). The first line goes from left-parenthesis negative 3 comma negative 3 right-parenthesis to left-parenthesis 0 comma 3 right-parenthesis. The second line goes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma negative 2 right-parenthesis. The coordinate plane is shaded above the first line and to the left of the second line. The first line goes from left-parenthesis negative 3 comma negative 3 right-parenthesis to left-parenthesis 0 comma 3 right-parenthesis. The second line goes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma negative 2 right-parenthesis. The coordinate plane is shaded above the first line and to the right of the second line. Image with alt text: A coordinate plane is shown with two solid boundary lines graphed with their intersection in the second quadrant. Overlapping shaded region includes the points (negative 2, 5), (negative 1, 2), and (0, 4). The first line goes from left-parenthesis negative 3 comma negative 3 right-parenthesis to left-parenthesis 0 comma 3 right-parenthesis. The second line goes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma negative 2 right-parenthesis. The coordinate plane is shaded above the first line and to the right of the second line.
The correct option is: "The first line goes from (-3,-3) to (0,3). The second line goes through the points (-2,4) and (0,-2). The coordinate plane is shaded below the first line and to the left of the second line."
The correct graph representation of the system of inequalities y ≤ 2x + 3 and 3x + y ≤ –2 is:
Image with alt text: A coordinate plane is shown with two solid boundary lines graphed with their intersection in the second quadrant. Overlapping shaded region includes the points (negative 1, 0), (negative 2, negative 3), and (0, negative 5).
The first line goes from (-3, -3) to (0, 3). The second line goes through the points (-2, 4) and (0, -2). The coordinate plane is shaded below the first line and to the left of the second line.
The correct graph that represents the system of inequalities y ≤ 2x + 3 and 3x + y ≤ –2 is the one described in the first response.
To understand how to graph these inequalities, we need to analyze each inequality separately and then combine their graphs to find the overlapping region, which represents the solution to the system.
1. Let's graph the first inequality, y ≤ 2x + 3:
- To plot this equation, start by graphing the line y = 2x + 3. Identify two points on the line, such as (-3, -3) and (0, 3) mentioned in the first option.
- Draw a solid line through these two points.
- Since the inequality is "y ≤ 2x + 3," we need to shade the region below the line.
2. Now, let's graph the second inequality, 3x + y ≤ -2:
- To plot this equation, rearrange it as y ≤ -3x - 2 to bring it to slope-intercept form.
- Identify two points on this line, such as (-2, 4) and (0, -2) mentioned in the first option.
- Draw a solid line through these two points.
- Since the inequality is "3x + y ≤ -2," we need to shade the region to the left of the line.
3. Lastly, locate the overlapping region of the shaded regions from both inequalities.
- This region represents the solution to the system of inequalities.
- The points mentioned in the first option, such as (-1, 0), (-2, -3), and (0, -5), lie within this overlapping region.
Therefore, the correct graph representing the system of inequalities is the one described in the first response.