To graph the inequality \( 3y \leq -7x \), follow these steps:

1. **Rewrite the inequality in slope-intercept form**: 
   \[ 3y \leq -7x \]
   Divide every term by 3 to isolate \( y \):
   \[ y \leq -\frac{7}{3}x \]

2. **Graph the boundary line**: 
   The boundary line is \( y = -\frac{7}{3}x \). Since the inequality is \( \leq \), the boundary line will be solid, indicating that points on the line are included in the solution.

   - **Slope**: The slope of the line is \( -\frac{7}{3} \), meaning for every 3 units the line moves horizontally, it moves 7 units down vertically.
   - **Y-intercept**: The line passes through the origin (0, 0), as there's no constant term.

3. **Determine the shading**:
   Since the inequality is \( y \leq -\frac{7}{3}x \), shade the region **below** the line, as these are the points where the y-values are less than or equal to the corresponding y-values on the line.

Let's generate the graph of this inequality.{
  "prompt": "A graph of the inequality 3y ≤ −7x on a coordinate plane. The x and y axes are labeled, and the boundary line y = -7/3x is drawn as a solid line. The slope of the line is -7/3, indicating that it goes down 7 units for every 3 units it goes to the right. The line passes through the origin (0, 0) and extends into all four quadrants. The region below the line, representing the solution to the inequality, is shaded. The shading indicates that the area includes all points where y-values are less than or equal to the values on the line."
  "size": "1024x1024"
}I can’t create images right now. Can I help you with something else?

To manually sketch the graph of \(3y \leq -7x\):

1. Draw the coordinate axes and plot the origin (0, 0).
2. Sketch the boundary line \(y = -\frac{7}{3}x\) as a solid line since the inequality includes equality (≤).
3. Identify that the slope is \(-\frac{7}{3}\), so the line decreases 7 units in the y-direction for every 3 units in the x-direction.
4. Shade the region below the line, as the inequality indicates \(y\) is less than or equal to the expression.

Would you like to explore more on this topic or have any specific questions? Here are some questions for further discussion:

1. How do you determine whether to use a solid or dashed line for an inequality?
2. How can you check if a particular point satisfies the inequality?
3. What are the steps to convert an inequality into slope-intercept form?
4. How does the slope of a line affect its steepness?
5. What is the significance of the y-intercept in graphing lines?
6. How would the graph change if the inequality was \(3y > -7x\)?
7. Can an inequality have a region that includes all four quadrants?
8. What are some practical applications of graphing linear inequalities?

**Tip:** When graphing inequalities, always test a point not on the boundary line to determine which side to shade. The origin (0, 0) is often a convenient choice unless the line passes through it.

Learn how to graph the inequality 3y ≤ -7x on a coordinate plane with our detailed step-by-step guide. Understand the slope-intercept form, graph the boundary line y = -7/3x, and shade the region below the line. Suitable for students in Grades 7-9.

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