Question
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Graph the inequality.
y<−x+3
Answers
Answered by
GPT-4o mini
To graph the inequality \( y < -x + 3 \), follow these steps:
1. **Graph the Boundary Line**:
- First, graph the line \( y = -x + 3 \). This line will be the boundary for the inequality.
- To do this, identify two points on the line:
- When \( x = 0 \):
\[
y = -0 + 3 = 3 \quad \text{(Point: (0, 3))}
\]
- When \( y = 0 \):
\[
0 = -x + 3 \implies x = 3 \quad \text{(Point: (3, 0))}
\]
- Plot these points on a Cartesian plane: (0, 3) and (3, 0).
- Since the inequality is \( < \), draw a dashed line through these points. This indicates that points on the line itself are not included in the solution (as denoted by the absence of “equal” in the inequality).
2. **Shade the Region**:
- Since the inequality is \( y < -x + 3 \), you will need to shade the region below the line.
- To determine which side to shade, you can pick a test point that is not on the line, such as (0, 0):
- Substitute (0, 0) into the inequality:
\[
0 < -0 + 3 \implies 0 < 3 \quad \text{(True)}
\]
- Since the test point (0, 0) satisfies the inequality, shade the area below the line.
Your final graph will have a dashed line passing through the points (0, 3) and (3, 0), and the area below the line will be shaded, representing all the points (x, y) that satisfy the inequality \( y < -x + 3 \).
1. **Graph the Boundary Line**:
- First, graph the line \( y = -x + 3 \). This line will be the boundary for the inequality.
- To do this, identify two points on the line:
- When \( x = 0 \):
\[
y = -0 + 3 = 3 \quad \text{(Point: (0, 3))}
\]
- When \( y = 0 \):
\[
0 = -x + 3 \implies x = 3 \quad \text{(Point: (3, 0))}
\]
- Plot these points on a Cartesian plane: (0, 3) and (3, 0).
- Since the inequality is \( < \), draw a dashed line through these points. This indicates that points on the line itself are not included in the solution (as denoted by the absence of “equal” in the inequality).
2. **Shade the Region**:
- Since the inequality is \( y < -x + 3 \), you will need to shade the region below the line.
- To determine which side to shade, you can pick a test point that is not on the line, such as (0, 0):
- Substitute (0, 0) into the inequality:
\[
0 < -0 + 3 \implies 0 < 3 \quad \text{(True)}
\]
- Since the test point (0, 0) satisfies the inequality, shade the area below the line.
Your final graph will have a dashed line passing through the points (0, 3) and (3, 0), and the area below the line will be shaded, representing all the points (x, y) that satisfy the inequality \( y < -x + 3 \).