Question
Give me a step by step how to graph y < 5 and y > 2x and where do I plot the points
Answers
GPT-4o mini
To graph the inequalities \( y < 5 \) and \( y > 2x \), follow these steps:
### Step 1: Graph the Line \( y = 5 \)
1. **Draw the line**: This line is horizontal and runs through all points where \( y \) is equal to 5.
2. **Use a dashed line**: Since the inequality is \( y < 5 \) (not \( y \leq 5 \)), draw the line dashed to indicate that points on the line are not included in the solution.
### Step 2: Graph the Line \( y = 2x \)
1. **Find points to plot**: To graph the line \( y = 2x \), you can find points by plugging in values for \( x \):
- If \( x = 0 \), then \( y = 2(0) = 0 \) → point (0, 0).
- If \( x = 1 \), then \( y = 2(1) = 2 \) → point (1, 2).
- If \( x = -1 \), then \( y = 2(-1) = -2 \) → point (-1, -2).
2. **Draw the line**: Connect the points you plotted to draw the line.
3. **Use a dashed line**: Since the inequality is \( y > 2x \) (not \( y \geq 2x \)), draw the line dashed.
### Step 3: Determine the Region for Each Inequality
1. **For \( y < 5 \)**: Shade the area **below the line** \( y = 5 \).
2. **For \( y > 2x \)**: Shade the area **above the line** \( y = 2x \).
### Step 4: Identify the Solution Region
The solution to the system of inequalities is where the shaded regions overlap. To find this area, check the following:
- Above the \( y = 2x \) line and below the \( y = 5 \) line.
### Example Points to Plot
For checking points, consider:
1. **Test a point inside the overlapping region, e.g., (1, 4)**:
- \( 4 < 5 \) → True (satisfies \( y < 5 \))
- \( 4 > 2(1) = 2 \) → True (satisfies \( y > 2x \))
2. **Test a point outside the region, e.g., (0, 3)**:
- \( 3 < 5 \) → True (satisfies \( y < 5 \))
- \( 3 > 2(0) = 0 \) → True (satisfies \( y > 2x \))
- But (0, 3) is not above \( y = 2x\).
By following these steps, you can successfully graph the inequalities and determine the solution region. The final graph will show two dashed lines and the appropriate shaded areas based on the inequalities provided.
### Step 1: Graph the Line \( y = 5 \)
1. **Draw the line**: This line is horizontal and runs through all points where \( y \) is equal to 5.
2. **Use a dashed line**: Since the inequality is \( y < 5 \) (not \( y \leq 5 \)), draw the line dashed to indicate that points on the line are not included in the solution.
### Step 2: Graph the Line \( y = 2x \)
1. **Find points to plot**: To graph the line \( y = 2x \), you can find points by plugging in values for \( x \):
- If \( x = 0 \), then \( y = 2(0) = 0 \) → point (0, 0).
- If \( x = 1 \), then \( y = 2(1) = 2 \) → point (1, 2).
- If \( x = -1 \), then \( y = 2(-1) = -2 \) → point (-1, -2).
2. **Draw the line**: Connect the points you plotted to draw the line.
3. **Use a dashed line**: Since the inequality is \( y > 2x \) (not \( y \geq 2x \)), draw the line dashed.
### Step 3: Determine the Region for Each Inequality
1. **For \( y < 5 \)**: Shade the area **below the line** \( y = 5 \).
2. **For \( y > 2x \)**: Shade the area **above the line** \( y = 2x \).
### Step 4: Identify the Solution Region
The solution to the system of inequalities is where the shaded regions overlap. To find this area, check the following:
- Above the \( y = 2x \) line and below the \( y = 5 \) line.
### Example Points to Plot
For checking points, consider:
1. **Test a point inside the overlapping region, e.g., (1, 4)**:
- \( 4 < 5 \) → True (satisfies \( y < 5 \))
- \( 4 > 2(1) = 2 \) → True (satisfies \( y > 2x \))
2. **Test a point outside the region, e.g., (0, 3)**:
- \( 3 < 5 \) → True (satisfies \( y < 5 \))
- \( 3 > 2(0) = 0 \) → True (satisfies \( y > 2x \))
- But (0, 3) is not above \( y = 2x\).
By following these steps, you can successfully graph the inequalities and determine the solution region. The final graph will show two dashed lines and the appropriate shaded areas based on the inequalities provided.