To determine which graph best represents the inequality $2x - y \leq 4$, let's go through the solution step-by-step.

Step 1: Rewrite the Inequality

The inequality $2x - y \leq 4$ can be rearranged into slope-intercept form $y \geq 2x - 4$, which is equivalent to: $y \leq 2x - 4$

Step 2: Interpret the Graph

1. Boundary Line: The inequality $y \leq 2x - 4$ has a boundary line at $y = 2x - 4$.

   • This line has a slope of 2 and a y-intercept at $-4$.
   • The line should be solid because the inequality is “$\leq$” (inclusive).

2. Shading: Since the inequality is $y \leq 2x - 4$, the graph should be shaded below the line, indicating all points where $y$ values are less than or equal to $2x - 4$.

Step 3: Analyze the Options

Based on these criteria:

• Graph A: Solid line, shaded below the line.
• Graph B: Incorrect shading (above the line).
• Graph C: Incorrect shading (above the line).
• Graph D: Solid line, shaded above the line.

Conclusion

Graph A is the correct representation of the inequality $2x - y \leq 4$ because it has:

• A solid boundary line on $y = 2x - 4$.
• Shading below the line.

Would you like further details on any part of this explanation, or have additional questions?

Here are five related questions to deepen understanding:

1. How do we determine where to shade on a graph for different inequality symbols?
2. Why is the boundary line solid instead of dashed in this inequality?
3. How would the graph change if the inequality were $2x - y > 4$?
4. How does the slope of the line affect the direction of shading in inequalities?
5. What is the significance of rewriting inequalities in slope-intercept form?

Tip: Always check the direction of the inequality to know where to shade, especially after rearranging it to slope-intercept form.