To determine which point is a solution to the inequality represented by the line segment between the points (-3, -1) and (3, 3), we first need to find the equation of the line defined by these points.
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Calculate the slope (m): The slope \( m \) between the points (-3, -1) and (3, 3) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - (-1)}{3 - (-3)} = \frac{4}{6} = \frac{2}{3} \]
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Use point-slope form to find the equation of the line: Using the point (-3, -1) and the slope \( \frac{2}{3} \): \[ y - y_1 = m(x - x_1) \] Substituting in, we have: \[ y - (-1) = \frac{2}{3}(x - (-3)) \] Simplifying this gives: \[ y + 1 = \frac{2}{3}(x + 3) \] Expanding this: \[ y + 1 = \frac{2}{3}x + 2 \ y = \frac{2}{3}x + 1 \]
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Determine the inequality: Since the graph shows a line segment from (-3, -1) to (3, 3), the solution to the inequality will likely either be above or below this line. Points below this line will satisfy \( y < \frac{2}{3}x + 1 \), while points above satisfy \( y > \frac{2}{3}x + 1 \).
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Testing points: Let's test a point below the line, like (0, 0): \[ 0 < \frac{2}{3}(0) + 1 \ 0 < 1 \quad \text{(True, so (0, 0) is a solution)} \]
Now, let’s test (2, 2): \[ 2 > \frac{2}{3}(2) + 1 \ 2 > \frac{4}{3} + 1 \ 2 > \frac{7}{3} \quad \text{(False, so (2, 2) is not a solution)} \]
Testing (3, 4): \[ 4 > \frac{2}{3}(3) + 1 \ 4 > 2 + 1 \ 4 > 3 \quad \text{(True, so (3, 4) is a solution)} \]
To summarize, points below the line segment are solutions to the inequality. If you're looking for a specific solution that falls on or below the segment, you'd say:
- Any points such as (0, 0), (1, 1), (2, 2), etc., below the line are valid solutions.
- Ensure to check if you're looking for specific regions or boundary inclusion/exclusion to determine the exact inequality.
If you have other points you want to check, feel free to share!
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