Question

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Explanation:

Identify boundary line properties

The visible graphs show a boundary line passing through specific grid intersections. Let's determine the equation of this boundary line.
Looking closely at the second graph (which is highlighted):

• The \(y\)-intercept is at \((0, 2)\).
• Another clear point on the line is \((1, -1)\) or \((2, -4)\).
• Let's calculate the slope \(m\):

$$ m = \frac{-1 - 2}{1 - 0} = -3 $$

• Thus, the equation of the boundary line is:

$$ y = -3x + 2 $$

Analyze inequality representations

In linear inequality graphing:

• A dashed boundary line represents a strict inequality (\(<\) or \(>\)).
• A solid boundary line represents a non-strict inequality (\(\le\) or \(\ge\)).
• The top graph shows a dashed boundary line:

$$ y < -3x + 2 \quad \text{or} \quad y > -3x + 2 $$

• The bottom graph shows a solid boundary line:

$$ y \le -3x + 2 \quad \text{or} \quad y \ge -3x + 2 $$

Determine shaded regions

Let's analyze the shaded region for the selected bottom graph:

• The shaded region lies to the left/below the solid line.
• Let's test the origin \((0,0)\), which is in the shaded region:

$$ 0 \le -3(0) + 2 \implies 0 \le 2 \quad (\text{True}) $$

• Therefore, the bottom graph represents the inequality:

$$ y \le -3x + 2 $$

• The top graph has a dashed line with the same shaded region, representing:

$$ y < -3x + 2 $$

Answer:

The highlighted bottom graph represents the linear inequality with a solid boundary line:

• The graph of \(y \le -3x + 2\) (solid boundary line with shading below/left) (Correct answer)
• The graph of \(y < -3x + 2\) (dashed boundary line with shading below/left)