To graph the inequality \(6x + y < -6\), follow these steps:
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Rewrite the inequality in slope-intercept form: Start by isolating \(y\): \[ y < -6x - 6 \] This is now in the form \(y < mx + b\), where \(m = -6\) (the slope) and \(b = -6\) (the y-intercept).
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Graph the boundary line: The boundary line can be found by graphing the equation \(6x + y = -6\) or \(y = -6x - 6\).
- To graph this line, identify two points.
- When \(x = 0\): \[ y = -6(0) - 6 = -6 \quad \text{(Point: (0, -6))} \]
- When \(x = -1\): \[ y = -6(-1) - 6 = 6 - 6 = 0 \quad \text{(Point: (-1, 0))} \]
- Plot these points: (0, -6) and (-1, 0).
- Draw a dashed line through these points. The dashed line indicates that points on the line are not included in the solution set, as the inequality is strict (\(<\)).
- To graph this line, identify two points.
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Determine which side to shade: Since the inequality is \(y < -6x - 6\), you need to shade the area below the line.
- A simple test point can help you determine which side to shade. A common choice is (0, 0): \[ 6(0) + 0 < -6 \quad \Rightarrow \quad 0 < -6 \quad \text{(This is false)} \] Since (0, 0) does not satisfy the inequality, shade the opposite side of the line from the point (0, 0).
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Final result: You should have a dashed line representing \(6x + y = -6\) and the area below the line shaded to represent the solution to the inequality \(6x + y < -6\).
Your graph should now accurately reflect the inequality! Here’s an outline of what it looks like:
- Dashed line through points (0, -6) and (-1, 0).
- Shaded area below the line.