choose the inequality that represents the following graph.
Boundary line equation
The boundary passes through the points (0, 4) and (2, 0), so the slope is \(m=\frac{0-4}{2-0}=-2\). The y-intercept occurs at \(x=0\), giving \(b=4\). The boundary line is \[ y=-2 \cdot x + 4. \]
Shaded side and equality
The shaded region lies above the boundary line. A point clearly in the shaded region is (0, 5). Substitution gives \(5\) compared with \(-2 \cdot 0 + 4 = 4\), so the inequality must use the “greater than” direction. The boundary is drawn as a solid line, so equality is included.
Answer: \(y \ge -2 \cdot x + 4\).
Equivalent algebraic forms
Several equivalent inequalities describe the same shaded half-plane:
- \(y \ge -2 \cdot x + 4\) (slope-intercept form).
- \(2 \cdot x + y \ge 4\) (standard form).
- \(y - 4 \ge -2 \cdot x\) (isolating the intercept).
Graph features and inequality symbols
| Graph feature | Meaning in the inequality |
|---|---|
| Solid boundary line | Boundary points are included; the symbol is \(\ge\) or \(\le\). |
| Dashed boundary line | Boundary points are excluded; the symbol is \(>\) or \(<\). |
| Shading above a line | \(y\) values exceed the boundary; the direction is \(y \ge\) or \(y >\). |
| Shading below a line | \(y\) values fall below the boundary; the direction is \(y \le\) or \(y <\). |
Common pitfalls
A negative slope can tempt an incorrect “above/below” interpretation when reading from left to right; the shading relative to the line (not the sign of the slope) governs the inequality direction. The solid versus dashed boundary controls whether equality is included.