To graph the inequality $y > 2 - 4x$, follow these steps:

1. Rewrite the inequality in slope-intercept form:

   • The inequality is already in slope-intercept form: $y > -4x + 2$.
   • Here, the slope ($m$) is $-4$, and the y-intercept ($b$) is $2$.

2. Graph the boundary line:

   • Start by graphing the line $y = -4x + 2$. Use a dashed line because the inequality is strict ($>$, not $\geq$).
   • Plot the y-intercept at $(0, 2)$.
   • Use the slope $-4$ to find another point: from $(0, 2)$, go down 4 units and right 1 unit to locate $(1, -2)$.

3. Shade the correct region:

   • Since the inequality is $y > -4x + 2$, shade the region above the dashed line. This represents all the points where $y$ is greater than $-4x + 2$.

Let me graph this for you. numpy as np import matplotlib.pyplot as plt

Define the line equation y = -4x + 2

x = np.linspace(-2, 2, 400) # Range of x values for the graph y = -4 * x + 2

Create the plot

plt.figure(figsize=(8, 6))

Plot the dashed boundary line

plt.plot(x, y, 'r--', label='y = -4x + 2 (Boundary Line)')

Shade the region above the line (inequality y > -4x + 2)

plt.fill_between(x, y, y2=10, where=(y < 10), color='orange', alpha=0.5, label='y > -4x + 2')

Add labels, title, and legend

plt.title("Graph of Inequality: y > -4x + 2") plt.xlabel("x") plt.ylabel("y") plt.axhline(0, color='black', linewidth=0.5) plt.axvline(0, color='black', linewidth=0.5) plt.grid(color='gray', linestyle='--', linewidth=0.5) plt.legend() plt.xlim(-2, 2) plt.ylim(-4, 6)

Show the plot

plt.show()Here is the graph of the inequality $y > 2 - 4x$. The dashed red line represents the boundary $y = 2 - 4x$, and the shaded orange region indicates where $y$ is greater than $2 - 4x$.

Do you have any questions, or would you like more details about this process?

Follow-up Questions:

1. What is the significance of using a dashed line in the graph?
2. How can you determine which region to shade for inequalities like $y > -4x + 2$?
3. Can you explain how to graph the inequality $y \geq 2 - 4x$ instead?
4. How does the slope $-4$ affect the direction of the boundary line?
5. How would the graph change if the inequality were $y < -4x + 2$?

Tip:

When graphing linear inequalities, always test a point (like $(0,0)$) to verify which side of the line satisfies the inequality.