Question

how to graph the inequality 6x+y<-6

Answer 1

To graph the inequality \(6x + y < -6\), follow these steps:

1. **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).

2. **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 (\(<\)).

3. **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).

4. **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.