Question

Solve the inequality 6x+3y > -27 then graph it.

Answer 1

To solve the inequality \( 6x + 3y > -27 \), we can start by rewriting it in slope-intercept form (i.e., \( y = mx + b \)), which makes it easier to understand and graph.

1. **Isolate \( y \)**:
\[
6x + 3y > -27
\]
Subtract \( 6x \) from both sides:
\[
3y > -6x - 27
\]
Now, divide every term by \( 3 \):
\[
y > -2x - 9
\]

This inequality indicates the region above the line \( y = -2x - 9 \).

2. **Graphing the line**:
- First, graph the line \( y = -2x - 9 \). This is a straight line with a slope of -2 and a y-intercept of -9.
- To find some points, you can plug in values for \( x \):
- For \( x = 0 \):
\[
y = -2(0) - 9 = -9 \quad \text{(point: (0, -9))}
\]
- For \( x = -2 \):
\[
y = -2(-2) - 9 = 4 - 9 = -5 \quad \text{(point: (-2, -5))}
\]
- For \( x = 1 \):
\[
y = -2(1) - 9 = -2 - 9 = -11 \quad \text{(point: (1, -11))}
\]

Plot these points and draw a dashed line through them, as the inequality is strict (\( > \), not \( \geq \)) indicating that the points on the line are not included in the solution.

3. **Shading**:
Since the inequality is \( y > -2x - 9 \), shade the region above the line.

4. **Final Graph**:
- X-axis: Horizontal line.
- Y-axis: Vertical line.
- Draw the dashed line \( y = -2x - 9 \).
- Shade the region above the line.

This represents the solution to the inequality \( 6x + 3y > -27 \).

If you have graphing software or a graphing calculator, you can enter the line equation and adjust the shading accordingly.