To determine the inequality represented by the graph with endpoints (3, 2) and (-3, -6), we can first find the slope of the line connecting these points and then determine the inequality based on the orientation of the line and the region it represents.
-
Find the slope (m) between the two points (3, 2) and (-3, -6):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 2}{-3 - 3} = \frac{-8}{-6} = \frac{4}{3} \]
-
Use the point-slope form to find the equation of the line passing through one of the points. Using point (3, 2):
\[ y - 2 = \frac{4}{3}(x - 3) \]
Simplifying further:
\[ y - 2 = \frac{4}{3}x - 4 \] \[ y = \frac{4}{3}x - 2 \]
-
The above equation, \(y = \frac{4}{3}x - 2\), represents the line that separates two regions in the coordinate plane.
-
Determine the direction of the inequality. We need to check which side of the line the solution lies. If the line is drawn solid, it indicates "≤" or "≥". If dashed, it indicates "<" or ">". Since the exact graph hasn't been provided, let's say we observe the graph, and it indicates that the region above the line is shaded, implying our inequality would be \(y > \frac{4}{3}x - 2\).
-
Conversely, if the shaded area were below the line, it would be \(y < \frac{4}{3}x - 2\).
Thus, based on the graph, you would identify the inequality as either \(y < \frac{4}{3}x - 2\) or \(y > \frac{4}{3}x - 2\), depending on which side of the line is shaded.
1 answer