To represent the solution to the system of inequalities you've given:
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First Inequality: \[ -3x + y \geq -4 \] This can be rearranged to express \(y\) in terms of \(x\): \[ y \geq 3x - 4 \]
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Second Inequality: \[ y < -2x + 3 \]
Now we have the inequalities explicitly in terms of \(y\):
- \(y \geq 3x - 4\) (the region above or on the line \(y = 3x - 4\))
- \(y < -2x + 3\) (the region below the line \(y = -2x + 3\))
Graphical Representation
To graph the system of inequalities:
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For the first inequality (\(y \geq 3x - 4\)):
- The line \(y = 3x - 4\) will be drawn with a solid line (because it's a "greater than or equal to" inequality), and the area above the line will be shaded.
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For the second inequality (\(y < -2x + 3\)):
- The line \(y = -2x + 3\) will be drawn with a dashed line (because it's a "less than" inequality), and the area below the line will be shaded.
Solution Area
The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap.
Thus, to summarize the original inequalities, the overall representation of the solution is:
- The area above or on the line \(y = 3x - 4\) (solid line) AND below the line \(y = -2x + 3\) (dashed line).
To write this in a concise form, the solution region can be expressed by the following: \[ 3x - 4 \leq y < -2x + 3 \]
This final expression indicates that \(y\) must satisfy both conditions from the original inequalities.