Which linear inequality is represented by the graph?
The Correct Answer and Explanation is:
ChatGPT said:
The graph shows a shaded region above a solid straight line, which represents the solution set of a linear inequality.
Correct Linear Inequality:
y≥x+2y \geq x + 2y≥x+2
Explanation
To determine the correct linear inequality represented by the graph, we follow these steps:
- Identify the boundary line:
The graph shows a straight line that intersects the y-axis at (0, 2) and goes through the point (−2, 0).
Using these two points, we can find the slope of the line: slope=0−2−2−0=−2−2=1\text{slope} = \frac{0 – 2}{-2 – 0} = \frac{-2}{-2} = 1slope=−2−00−2=−2−2=1 So, the equation of the line is: y=x+2y = x + 2y=x+2 - Determine the type of inequality:
The line is solid, which means points on the line are included in the solution. This tells us the inequality symbol is “≥” or “≤” rather than a strict inequality like “>” or “<”. - Identify the shaded region:
The shaded region is above the line. In the coordinate plane, this means that the values of y are greater than the values given by the line equation. Therefore, the inequality is: y≥x+2y \geq x + 2y≥x+2 - Check a test point:
To verify, we can test the point (0, 4), which lies in the shaded region: 4≥0+2⇒4≥2 ✓ True4 \geq 0 + 2 \Rightarrow 4 \geq 2 \text{ ✓ True}4≥0+2⇒4≥2 ✓ True So, the inequality holds in the shaded region.
Conclusion:
The shaded region and boundary line correspond to the inequality:y≥x+2\boxed{y \geq x + 2}y≥x+2
This inequality includes all points on and above the line y=x+2y = x + 2y=x+2.
