Which linear inequality is represented by the graph?
y < 2/3x + 3 y >3/2 x + 3
y >2/3 x + 3
y < 3/2x + 3
The Correct Answer and Explanation is :
The correct answer is: (y < \frac{3}{2}x + 3)
To determine which linear inequality is represented by the provided graph, we first need to analyze the characteristics of the graph based on the equations given. The graph in the image displays a line and shading that indicates the solution set for an inequality.
- Identify the line: The line shown appears to be the boundary for the inequality. The equation of the line can be derived from the slope-intercept form (y = mx + b), where (m) is the slope and (b) is the y-intercept.
- Determine the slope and intercept: In the given options, the slopes of the lines are (2/3) and (3/2). The y-intercept appears to be (3). A line with a slope of (2/3) rises gradually, while a slope of (3/2) is steeper. You can check the positioning of the line relative to the given y-intercepts to confirm which one matches the graph.
- Shading direction: The shading direction indicates the solutions to the inequality. If the area above the line is shaded, it represents a “greater than” inequality (either (y > mx + b) or (y \geq mx + b)). Conversely, if the area below the line is shaded, it represents a “less than” inequality (either (y < mx + b) or (y \leq mx + b)).
- Solid vs. dashed line: If the line is solid, the inequality includes equal to (≥ or ≤), while a dashed line indicates that it does not include equal to (> or <).
Now, based on these observations, let’s summarize:
- The graph likely has a boundary line derived from one of the equations given.
- If the shading is below the line and the line is dashed, then the correct inequality is of the form (y < mx + b).
Considering the provided options, if the line is indeed dashed and shading is below the line, the correct inequality is (y < \frac{3}{2}x + 3), as it matches the dashed line trend with the corresponding slope of (3/2) and y-intercept of (3).