The shaded region shown represents the solutions to which inequality?

1. TRANSLATE the graph to identify the line equation

First, we need to find the equation of the boundary line shown in black.

Given information from the graph:

• The line passes through \((0, 2)\) - this is where the line crosses the y-axis
• The line passes through \((2, 8)\) - another clearly visible point

2. Calculate the slope using the two points

Using our two points \((0, 2)\) and \((2, 8)\):

\(\mathrm{Slope = m = \frac{8 - 2}{2 - 0}}\)

\(\mathrm{= \frac{6}{2}}\)

\(\mathrm{= 3}\)

3. INFER the complete line equation

Since we know:

• Slope \(\mathrm{m = 3}\)
• Y-intercept \(\mathrm{b = 2}\) (the line crosses the y-axis at y = 2)

The line equation is: \(\mathrm{y = 3x + 2}\)

4. TRANSLATE the shaded region to determine the inequality

Now we need to figure out which inequality symbol to use.

Key observation from the graph:

• The gray shaded region is below the line
• "Below the line" means for any given x-value, the y-values in the shaded region are less than the y-value on the line

5. INFER the correct inequality symbol

Since the shaded region is below the line, and below means "less than":

• The inequality is: \(\mathrm{y \lt 3x + 2}\)

Looking at the answer choices, this is Choice C.

Answer: C (\(\mathrm{y \lt 3x + 2}\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing the direction of the inequality

Students often mix up which inequality symbol corresponds to which region:

• They see "below the line" but incorrectly think this means \(\mathrm{y \gt 3x + 2}\)
• Or they correctly identify "less than" but then doubt themselves and switch to "greater than"

This conceptual confusion about the relationship between spatial position (above/below) and mathematical symbols () leads them to select Choice A (\(\mathrm{y \gt 3x + 2}\)) instead of the correct answer.

Second Most Common Error:

Poor TRANSLATE skill: Misreading points on the graph

Students may:

• Incorrectly identify the y-intercept (reading it as -2 instead of +2)
• Miscalculate the slope by choosing wrong points or making arithmetic errors

If they read the y-intercept as -2 instead of +2, they get the equation \(\mathrm{y = 3x - 2}\), and if they also get the inequality direction correct, this leads them to select Choice D (\(\mathrm{y \lt 3x - 2}\)).

The Bottom Line:

This problem tests two distinct skills: (1) finding a line equation from its graph, and (2) connecting the spatial concept of "region below a line" with the mathematical symbol "<". Many students can do step 1 but struggle with the visual-to-symbolic translation in step 2, especially under time pressure when they second-guess the inequality direction.