The graph of the inequality `yle2x` and include all the points in which quadrant?

Options

A. Quadrant I
B. Quadrant II
C. Quadrant III
D. Quadrant IV

Answer

To solve the inequality \( y \leq 2x \) and determine which quadrant(s) it includes, we can follow these steps:

### Step 1: Understand the Inequality
The inequality \( y \leq 2x \) means we are looking for all points where the \( y \)-coordinate is less than or equal to \( 2x \).

### Step 2: Convert Inequality to Equation
To analyze the inequality, we first consider the corresponding equation:
\[ y = 2x \]
This equation represents a straight line.

### Step 3: Identify Points on the Line
We can find two points on the line for plotting:
- When \( x = 0 \):
\[ y = 2(0) = 0 \quad \Rightarrow \quad (0, 0) \]
- When \( x = 1 \):
\[ y = 2(1) = 2 \quad \Rightarrow \quad (1, 2) \]

### Step 4: Plot the Points and Draw the Line
Now, we plot the points \( (0, 0) \) and \( (1, 2) \) on the Cartesian plane and draw the line that passes through these points.

### Step 5: Determine the Region for the Inequality
Since the inequality is \( y \leq 2x \), we need to shade the region below the line \( y = 2x \) (including the line itself).

### Step 6: Test a Point to Determine the Valid Region
To confirm which side of the line satisfies the inequality, we can test a point. Let's test the point \( (1, -1) \):
\[ y = -1 \quad \text{and} \quad 2x = 2(1) = 2 \]
We check:
\[ -1 \leq 2 \quad \text{(True)} \]
This point satisfies the inequality, indicating that the region below the line is valid.

### Step 7: Identify the Quadrants
Now we analyze which quadrants are included in the shaded region:
- **Quadrant I**: \( x > 0, y > 0 \)
- **Quadrant II**: \( x < 0, y > 0 \)
- **Quadrant III**: \( x < 0, y < 0 \)
- **Quadrant IV**: \( x > 0, y < 0 \)

From the graph, we can see that the shaded region (where \( y \leq 2x \)) includes:
- Quadrant IV (where \( x > 0 \) and \( y < 0 \))
- Part of Quadrant III (where \( x < 0 \) and \( y < 0 \))
- Part of Quadrant II (where \( x < 0 \) and \( y > 0 \))

### Conclusion
The graph of the inequality \( y \leq 2x \) includes all points in Quadrant IV, as well as parts of Quadrants II and III. However, since the question specifically asks for which quadrant(s) it includes, the answer is:
**Quadrant IV.**