To interpret the information you've provided about the two lines and to answer the questions accurately, we'll need to determine the equations of the lines and their corresponding inequalities.
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Green Line: It is described as a dotted line passing through the points (-3, 0) and (0, 3). Since it's a dotted line, it indicates that the inequality is strict (not including the boundary line).
- The slope (m) of the line can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{0 - (-3)} = \frac{3}{3} = 1 \]
- The line equation can be written in slope-intercept form \(y = mx + b\), which gives: \[ y = x + 3 \]
Since the line is dotted, the inequality symbol for the green line is: \[ y < x + 3 \text{ or } y > x + 3 \] Depending on the region being shaded (below or above the line), you would specify which one it is. Typically, it is shaded below for a line with a positive slope.
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Purple Line: It is a solid line passing through the points (-3, 0) and (0, -3). Since it's a solid line, it indicates that the inequality includes the boundary line.
- The slope (m) of the purple line can be calculated similarly: \[ m = \frac{-3 - 0}{0 - (-3)} = \frac{-3}{3} = -1 \]
- The equation, in slope-intercept form, would be: \[ y = -x - 3 \]
As the line is solid, the inequality symbol for the purple line is: \[ y \leq -x - 3 \text{ or } y \geq -x - 3 \] Again, depending on the shading above or below the line, you would specify which one applies.
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Checking if the point (-3, -5) is a solution:
- For the green line inequality (\(y < x + 3\)): \[ -5 < -3 + 3 \implies -5 < 0 \text{ (TRUE)} \]
- For the purple line inequality (\(y \leq -x - 3\)): \[ -5 \leq -(-3) - 3 \implies -5 \leq 0 - 3 \implies -5 \leq -3 \text{ (FALSE)} \]
Since it does not satisfy both inequalities, (-3, -5) is not a solution to the system of inequalities.
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Checking if the point (1, -1) is a solution:
- For the green line inequality (\(y < x + 3\)): \[ -1 < 1 + 3 \implies -1 < 4 \text{ (TRUE)} \]
- For the purple line inequality (\(y \leq -x - 3\)): \[ -1 \leq -1 - 3 \implies -1 \leq -4 \text{ (FALSE)} \]
Since it does not satisfy both inequalities, (1, -1) is not a solution to the system of inequalities.
In summary:
- The inequality symbol for the green line is \(y < x + 3\).
- The inequality symbol for the purple line is \(y \leq -x - 3\).
- The point (-3, -5) is not a solution.
- The point (1, -1) is not a solution.
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