Question

Which point is a solution to the inequality shown in this graph? (Line, (0, 1) to (3, -1))

Answer 1

To determine which point is a solution to the inequality represented by the graph of the line from (0, 1) to (3, -1), we first need to establish the equation of the line.

1. **Find the slope (m)**:
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{3 - 0} = \frac{-2}{3}
\]

2. **Use the point-slope form of the equation**:
Using the point (0, 1):
\[
y - 1 = -\frac{2}{3}(x - 0)
\]
Simplifying gives:
\[
y = -\frac{2}{3}x + 1
\]

3. **Identify the inequality**:
Since the line endpoints suggest a negative slope and the y-intercept is positive, the region below this line is likely the solution set if the line is dashed (indicating that points on the line are not included in the solution). Thus, the inequality can be represented as:
\[
y < -\frac{2}{3}x + 1
\]
(if the line is solid, it would be \(y \leq -\frac{2}{3}x + 1\)).

4. **Check sample points**:
To check which points are solutions, substitute them into the inequality. You need to ensure the y-coordinate is less than the y-value on the line at that x-coordinate.

To verify a specific point, you can substitute its coordinates into the inequality. Let's assume a few sample points:

- Point A (0, 0):
\[
0 < -\frac{2}{3}(0) + 1 \quad \text{(true)}
\]
This point is a solution.

- Point B (1, 0):
\[
0 < -\frac{2}{3}(1) + 1 \Rightarrow 0 < -\frac{2}{3} + 1 \Rightarrow 0 < \frac{1}{3} \quad \text{(true)}
\]
This point is also a solution.

- Point C (2, 0):
\[
0 < -\frac{2}{3}(2) + 1 \Rightarrow 0 < -\frac{4}{3} + 1 \Rightarrow 0 < -\frac{1}{3} \quad \text{(false)}
\]
This point is not a solution.

The specific points that you check will give you valid solutions based on the inequality. Please provide specific points if you want to check for their validity against the established inequality!