To find the points that are solutions for the simultaneous inequalities \( y > 4x - 2 \) and \( y \geq -13x + 2 \), we need to identify points that fall above the line defined by \( y = 4x - 2 \) and on or above the line defined by \( y = -13x + 2 \).
Since I do not have access to the image of the graph, I cannot directly calculate the exact point you are asking about. However, here's how you can find the points that fulfill both inequalities:
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Plot the first line \( y = 4x - 2 \): This line has a slope of 4, which means it rises steeply. The y-intercept is at (0, -2). The region of interest will be above this line.
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Plot the second line \( y = -13x + 2 \): This line has a slope of -13, indicating a steep decline. The y-intercept is at (0, 2). The region of interest will include this line and everything above it.
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Find the intersection of these lines: Set the equations equal to each other to find any intersection points: \[ 4x - 2 = -13x + 2 \] Solve for \( x \):
\[ 4x + 13x = 2 + 2 \] \[ 17x = 4 \quad \Rightarrow \quad x = \frac{4}{17} \]
Substitute \( x \) back into either equation to find \( y \): \[ y = 4\left(\frac{4}{17}\right) - 2 = \frac{16}{17} - \frac{34}{17} = \frac{-18}{17} \]
Thus, the intersection point is \( \left( \frac{4}{17}, \frac{-18}{17} \right) \).
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Select points in the graph: Choose a point above the line \( y = 4x - 2 \) and above or on the line \( y = -13x + 2 \).
Once you identify points from the graph, make sure they satisfy the conditions of being above the first inequality and on or above the second inequality.
If you can provide the specific coordinates of points from the graph, I can help confirm or calculate whether they satisfy the inequalities.
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