To graph the inequality \(6x + y < 6\), follow these steps:
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Convert the inequality to an equation: Start by treating the inequality as an equation to find the boundary line. Set the inequality to an equation: \[ 6x + y = 6 \]
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Find the intercepts:
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Y-intercept: Set \(x = 0\) in the equation: \[ 6(0) + y = 6 \implies y = 6 \] So the y-intercept is the point \((0, 6)\).
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X-intercept: Set \(y = 0\) in the equation: \[ 6x + 0 = 6 \implies x = 1 \] So the x-intercept is the point \((1, 0)\).
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Plot the boundary line: Since the original inequality is "<", the boundary will not be included in the solution. Therefore, draw a dashed line through the points \((0, 6)\) and \((1, 0)\).
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Choose a test point: To determine which side of the line to shade, choose a point that is not on the boundary line, such as the origin \((0, 0)\): \[ 6(0) + 0 < 6 \implies 0 < 6 \quad \text{(True)} \] Since the test point satisfies the inequality, we shade the region that contains \((0, 0)\).
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Conclusion: The solution to the inequality \(6x + y < 6\) is the shaded area below the dashed line \(6x + y = 6\), including all points where \(y\) is less than the value given by the equation on the left side.
Summary:
When graphing the inequality \(6x + y < 6\):
- Draw a dashed line for \(6x + y = 6\).
- Shade the region below the line to represent the solution set where \(y < 6 - 6x\).