Question

how to graph the inequality 6x+y<6

Answer 1

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

1. **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:
\[
6x + y = 6
\]

2. **Find the intercepts:**
- **Y-intercept:** Set \(x = 0\) in the equation:
\[
6(0) + y = 6 \implies y = 6
\]
So the y-intercept is the point \((0, 6)\).

- **X-intercept:** Set \(y = 0\) in the equation:
\[
6x + 0 = 6 \implies x = 1
\]
So the x-intercept is the point \((1, 0)\).

3. **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 \((1, 0)\).

4. **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 \((0, 0)\).

5. **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 \(y < 6 - 6x\).