Question
Graph the inequality.
`y\ge x-3`
is it a dashed line? turn on shading or off? what are the corridnates please be sepsfic and add a answer summary
Answers
GPT-4o mini
To graph the inequality \( y \ge x - 3 \), let's break down the process step-by-step.
### Step 1: Graph the boundary line
1. **Convert the inequality to an equation**: The boundary line is represented by the equation \( y = x - 3 \).
2. **Identify two points to draw the line**:
- When \( x = 0 \):
\[
y = 0 - 3 = -3 \implies (0, -3)
\]
- When \( x = 3 \):
\[
y = 3 - 3 = 0 \implies (3, 0)
\]
3. **Plot the points (0, -3) and (3, 0)** on the graph.
4. **Draw the boundary line**: Since the inequality is \( y \ge x - 3 \), you will draw a solid line for the boundary because the inequality includes equality (≥).
### Step 2: Determine the shading
1. **Choose a test point** to determine where to shade. A common choice is (0, 0):
- Substitute \( (0, 0) \) into the inequality:
\[
0 \ge 0 - 3 \implies 0 \ge -3
\]
- Since this statement is true, shade the region that contains the point (0, 0).
### Step 3: Result Summary
1. **Line type**: Solid line (because of the ≥ sign).
2. **Shading**: Shade the region above and including the line \( y = x - 3 \) since the inequality \( y \ge x - 3 \) represents all points where \( y \) is greater than or equal to \( x - 3 \).
3. **Coordinates**:
- Sample points on the line: (0, -3) and (3, 0).
- Additional points that satisfy the inequality (for shading) could include (0, 0), (1, -1), and (2, -1).
### Final notes
- The graph effectively represents all points in the Cartesian plane such that the \( y \) value is greater than or equal to the value given by the line \( y = x - 3 \).
- Make sure to extend the shading infinitely in the direction that satisfies the inequality.
### Step 1: Graph the boundary line
1. **Convert the inequality to an equation**: The boundary line is represented by the equation \( y = x - 3 \).
2. **Identify two points to draw the line**:
- When \( x = 0 \):
\[
y = 0 - 3 = -3 \implies (0, -3)
\]
- When \( x = 3 \):
\[
y = 3 - 3 = 0 \implies (3, 0)
\]
3. **Plot the points (0, -3) and (3, 0)** on the graph.
4. **Draw the boundary line**: Since the inequality is \( y \ge x - 3 \), you will draw a solid line for the boundary because the inequality includes equality (≥).
### Step 2: Determine the shading
1. **Choose a test point** to determine where to shade. A common choice is (0, 0):
- Substitute \( (0, 0) \) into the inequality:
\[
0 \ge 0 - 3 \implies 0 \ge -3
\]
- Since this statement is true, shade the region that contains the point (0, 0).
### Step 3: Result Summary
1. **Line type**: Solid line (because of the ≥ sign).
2. **Shading**: Shade the region above and including the line \( y = x - 3 \) since the inequality \( y \ge x - 3 \) represents all points where \( y \) is greater than or equal to \( x - 3 \).
3. **Coordinates**:
- Sample points on the line: (0, -3) and (3, 0).
- Additional points that satisfy the inequality (for shading) could include (0, 0), (1, -1), and (2, -1).
### Final notes
- The graph effectively represents all points in the Cartesian plane such that the \( y \) value is greater than or equal to the value given by the line \( y = x - 3 \).
- Make sure to extend the shading infinitely in the direction that satisfies the inequality.