Question

This graph shows the solution of which inequality?
(3, 2) to (-3, -6)

Answer 1

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.

1. **Find the slope (m)** between the two points (3, 2) and (-3, -6):

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-6 - 2}{-3 - 3} = \frac{-8}{-6} = \frac{4}{3}
\]

2. **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:

\[
y - 2 = \frac{4}{3}x - 4
\]
\[
y = \frac{4}{3}x - 2
\]

3. The above equation, \(y = \frac{4}{3}x - 2\), represents the line that separates two regions in the coordinate plane.

4. **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 \(y > \frac{4}{3}x - 2\).

5. 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.