A specific graph is taken as the reference: a straight boundary line passes through the points (0, 3) and (1, 1), the boundary is drawn as a solid line, and the shaded region lies above the line. Those visual cues uniquely determine the inequality.
Algebra represented by the boundary line
The two labeled points on the boundary line are (0, 3) and (1, 1). The slope is \[ m=\frac{1-3}{1-0}=-2, \] and the y-intercept is \(3\), so the boundary line has equation \[ y=-2x+3. \]
Shading and inequality symbol
The shaded region lies above the line, so the y-values in the solution set are greater than the line’s y-values at the same x. The boundary is solid, so points on the line are included, giving a “greater than or equal to” symbol.
Boundary type and shading direction encode the inequality: a solid boundary corresponds to \( \ge \) or \( \le \), while a dashed boundary corresponds to \( > \) or \( < \). Shading above a line corresponds to \(y\) being greater than the boundary expression; shading below corresponds to \(y\) being less.
Answer among typical options
A standard multiple-choice framing for “which of the following inequalities matches the graph” is illustrated by the options below. The correct choice follows from the solid boundary and the shading above \(y=-2x+3\).
| Option | Inequality | Graph meaning |
|---|---|---|
| A | \(y \ge -2x + 3\) | Above the line, boundary included |
| B | \(y > -2x + 3\) | Above the line, boundary excluded (dashed) |
| C | \(y \le -2x + 3\) | Below the line, boundary included |
| D | \(y \ge 2x + 3\) | Above a line with positive slope |
Correct inequality: \(y \ge -2x + 3\).
Consistency checks
The test point (0, 4) lies in the shaded region and satisfies \(4 \ge -2\cdot 0 + 3\). The point (0, 2) lies below the boundary and fails \(2 \ge 3\), matching the unshaded region. Boundary points such as (0, 3) satisfy equality \(3 = -2\cdot 0 + 3\), consistent with the solid boundary.