Question
Can you explain step by step where to plot the points for y > 2x + 2 and x ≥ -3
Answers
GPT-4o mini
To plot the points that satisfy the inequalities \( y > 2x + 2 \) and \( x \geq -3 \), we can follow these steps:
### Step 1: Graph the Line for the Equality
Start by graphing the line for the equation \( y = 2x + 2 \):
1. **Find the y-intercept**: When \( x = 0 \), \( y = 2(0) + 2 = 2 \). So, the point (0, 2) is on the line.
2. **Find another point**: Choose another value for \( x \). For example, when \( x = 1 \):
- \( y = 2(1) + 2 = 4 \) leads to the point (1, 4).
3. **Plot these two points**: Plot (0, 2) and (1, 4) on the graph.
4. **Draw the Line**: Since the inequality is \( y > 2x + 2 \), draw a dashed line through these points to indicate that points on the line itself are not included in the solution (because it’s a strict inequality).
### Step 2: Shade the Area for \( y > 2x + 2 \)
Now, you need to shade the region above the line:
- To determine which side to shade, pick a test point that is not on the line. A common choice is (0, 0). Substitute into the inequality:
- \( 0 > 2(0) + 2 \) simplifies to \( 0 > 2 \), which is false.
- Since (0, 0) does not satisfy the inequality, shade the area above the line (where \( y \) values are greater than those on the line).
### Step 3: Graph the Vertical Line for \( x \geq -3 \)
Next, graph the vertical line for the inequality \( x = -3 \):
1. **Graphing the line**: Draw a vertical dashed line at \( x = -3 \). Since the inequality is \( x \geq -3 \), this line is a solid line, indicating that points on the line are included in the solution.
2. **Shade the right side**: Since the inequality is "greater than or equal to", shade the region to the right of this line. This includes all points where \( x \) values are greater than or equal to -3.
### Step 4: Identify the Intersection Area
Finally, the solution to the system of inequalities \( y > 2x + 2 \) and \( x \geq -3 \):
- Look at the area where the shading (above the line \( y = 2x + 2 \)) overlaps with the shading to the right of the line \( x = -3 \).
- This overlapping area represents all the points (x, y) that satisfy both inequalities.
### Summary
In summary, to plot the points for \( y > 2x + 2 \) and \( x \geq -3 \):
1. Draw a dashed line for \( y = 2x + 2 \) and shade above it.
2. Draw a solid vertical line for \( x = -3 \) and shade to the right.
3. The intersection of these shaded regions is the solution set.
### Step 1: Graph the Line for the Equality
Start by graphing the line for the equation \( y = 2x + 2 \):
1. **Find the y-intercept**: When \( x = 0 \), \( y = 2(0) + 2 = 2 \). So, the point (0, 2) is on the line.
2. **Find another point**: Choose another value for \( x \). For example, when \( x = 1 \):
- \( y = 2(1) + 2 = 4 \) leads to the point (1, 4).
3. **Plot these two points**: Plot (0, 2) and (1, 4) on the graph.
4. **Draw the Line**: Since the inequality is \( y > 2x + 2 \), draw a dashed line through these points to indicate that points on the line itself are not included in the solution (because it’s a strict inequality).
### Step 2: Shade the Area for \( y > 2x + 2 \)
Now, you need to shade the region above the line:
- To determine which side to shade, pick a test point that is not on the line. A common choice is (0, 0). Substitute into the inequality:
- \( 0 > 2(0) + 2 \) simplifies to \( 0 > 2 \), which is false.
- Since (0, 0) does not satisfy the inequality, shade the area above the line (where \( y \) values are greater than those on the line).
### Step 3: Graph the Vertical Line for \( x \geq -3 \)
Next, graph the vertical line for the inequality \( x = -3 \):
1. **Graphing the line**: Draw a vertical dashed line at \( x = -3 \). Since the inequality is \( x \geq -3 \), this line is a solid line, indicating that points on the line are included in the solution.
2. **Shade the right side**: Since the inequality is "greater than or equal to", shade the region to the right of this line. This includes all points where \( x \) values are greater than or equal to -3.
### Step 4: Identify the Intersection Area
Finally, the solution to the system of inequalities \( y > 2x + 2 \) and \( x \geq -3 \):
- Look at the area where the shading (above the line \( y = 2x + 2 \)) overlaps with the shading to the right of the line \( x = -3 \).
- This overlapping area represents all the points (x, y) that satisfy both inequalities.
### Summary
In summary, to plot the points for \( y > 2x + 2 \) and \( x \geq -3 \):
1. Draw a dashed line for \( y = 2x + 2 \) and shade above it.
2. Draw a solid vertical line for \( x = -3 \) and shade to the right.
3. The intersection of these shaded regions is the solution set.