Joel and Nolan are working on a math test, and both wrote a different inequality for the graph below. Which student is correct, and why?

Question

Joel
y ≤ 4x + 2
Nolan
y < 4x + 2
+ (0, 2)
서1, 2후
• Nolan, because the less then sign tells us that the line should be dashed/dotted and should be shaded below the line.
@ Nolan, because the less than sign tells us to shade above the line.
• Joel, because the less than tells us that the line is dashed/dotted and should be shoded below the line.
• Joel, becouse the less than or equal to symbol tells us that the line is solid and should be shaded below the line.

Answer 1

To determine which student is correct, let's analyze both inequalities in relation to the graph.

1. Understanding the inequalities:

   • Joel's inequality: \( y \leq 4x + 2 \)
   • Nolan's inequality: \( y < 4x + 2 \)

2. Interpreting the inequalities:

   • Joel's inequality (with the "≤" sign) indicates a solid line because the boundary line itself is included in the solution set. It also indicates that the region below this line should be shaded (including the line).
   • Nolan's inequality (with the "<" sign) indicates a dashed line because the boundary line itself is not included in the solution set. It also indicates that the region below the line should be shaded.

3. Standard rules for graphing inequalities:

   • If the inequality sign is "≤" or "≥", draw a solid line.
   • If the inequality sign is "<" or ">", draw a dashed line.
   • For inequalities where \( y < \) or \( y ≤ \), shade below the line.
   • For inequalities where \( y > \) or \( y ≥ \), shade above the line.

4. Conclusion based on the question options and the interpretation:

   • If the graph includes the line (solid line) and includes points on the line (like the point (0, 2)), then Joel is correct.
   • If the graph does not include the line (dashed line), then Nolan is correct.

Based on this reasoning, the correct answer aligns with the description of the inequalities and points mentioned in the question.

Therefore, if the graph shows a solid line and includes the point (0, 2), then Joel is correct because the inequality should be \( y \leq 4x + 2 \).

So you would choose:

• Joel, because the less than or equal to symbol tells us that the line is solid and should be shaded below the line.

If the graph is indeed represented with a dashed line, then Nolan would be the correct choice. Ensure you carefully check the graphical representation to clarify the correct inequality!