Question

which inequality is represented by the graph below? 1. $y > -3x + 2$ 2. $y geq -3x + 2$ 3. $y > -\frac{1}{2}x + 3$ 4. $y geq -\frac{1}{2}x + 3$

Explanation:

Step1: Analyze the line type (dashed or solid)

The graph has a dashed line, which means the inequality is strict (either \(>\) or \(<\)), so we can eliminate options with \(\geq\) or \(\leq\) (options 2 and 4).

Step2: Find the slope and y-intercept of the line

The line passes through \((0, 3)\) (y-intercept \(b = 3\)) and, for example, when \(y = 0\), \(x = 6\) (since from \((0,3)\) to \((6,0)\), the run is 6 and rise is -3). The slope \(m=\frac{\text{rise}}{\text{run}}=\frac{-3}{6}=-\frac{1}{2}\). So the equation of the line is \(y = -\frac{1}{2}x + 3\).

Step3: Determine the inequality direction

The shaded region is above the dashed line, so the inequality is \(y > -\frac{1}{2}x + 3\) (since above the line for a dashed line with negative slope, the inequality is \(y > mx + b\)).

Answer:

1. \(y > -\frac{1}{2}x + 3\)