To determine which point is a solution to the inequality represented by the line from (0, 1) to (3, -1), we first need to ascertain the equation of the line itself.
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Finding the slope (m) of the line:
- The coordinates of the points are (0, 1) and (3, -1).
- Slope \( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - 1}{3 - 0} = \frac{-2}{3} \).
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Finding the y-intercept (b) of the line:
- The y-intercept is obtained directly from the point (0, 1), so \( b = 1 \).
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Equation of the line:
- The equation can be expressed in slope-intercept form as: \[ y = -\frac{2}{3}x + 1 \]
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Understanding the inequality:
- The graph shows a line, and the area chosen must correspond to the inequality. Given that the line slopes downward from (0, 1) to (3, -1), it is likely that the area below the line is shaded, indicating a "less than" relationship (\( y < -\frac{2}{3}x + 1 \)).
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Testing the points:
- (0, 0): \[ 0 < -\frac{2}{3}(0) + 1 \implies 0 < 1 \quad \text{(True)} \]
- (0, -5): \[ -5 < -\frac{2}{3}(0) + 1 \implies -5 < 1 \quad \text{(True)} \]
- (5, -5): \[ -5 < -\frac{2}{3}(5) + 1 \implies -5 < -\frac{10}{3} + 1 \implies -5 < -\frac{10}{3} + \frac{3}{3} \implies -5 < -\frac{7}{3} \quad \text{(False)} \]
- (3, -1): \[ -1 < -\frac{2}{3}(3) + 1 \implies -1 < -2 + 1 \implies -1 < -1 \quad \text{(False)} \]
Therefore, the points that satisfy the inequality are (0, 0) and (0, -5).
Since both points are valid solutions, the question might be seeking only one solution. If forced to pick one, either could work, but usually, the point closer to the origin or simpler to evaluate is often preferred (in this case, (0, 0)).
So, a solution to the inequality shown in the graph is (0, 0).