What Are The Attributes Of The Boundary Line Of This Inequality?$-3x - 2y \ \textless \ 6$A. The Line Is Solid With A $y$-intercept At $(0, -2$\] And Slope Of $\frac{3}{2}$.B. The Line Is Solid With A

Understanding the Inequality

The given inequality is $-3x - 2y < 6$. To find the attributes of the boundary line, we need to first convert this inequality into the slope-intercept form, which is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Converting the Inequality to Slope-Intercept Form

To convert the inequality into the slope-intercept form, we need to isolate $y$ on one side of the inequality. We can do this by adding $3x$ to both sides of the inequality and then dividing both sides by $-2$.

$$-3x - 2y < 6$$

Adding $3x$ to both sides:

$$-2y < 3x + 6$$

Dividing both sides by $-2$:

$$y > -\frac{3}{2}x - 3$$

Identifying the Attributes of the Boundary Line

Now that we have the inequality in the slope-intercept form, we can identify the attributes of the boundary line.

Slope

The slope of the boundary line is the coefficient of $x$, which is $-\frac{3}{2}$.

$y$-Intercept

The $y$-intercept of the boundary line is the constant term, which is $-3$.

Conclusion

Based on the inequality $-3x - 2y < 6$, we can conclude that the attributes of the boundary line are:

• Slope: $\frac{3}{2}$
• $y$-Intercept: $-3$

The boundary line is a solid line because the inequality is strict (less than). The $y$-intercept is at $(0, -3)$, and the slope is $\frac{3}{2}$.

Comparison with the Given Options

Let's compare our findings with the given options:

A. The line is solid with a $y$-intercept at $(0, -2)$ and slope of $\frac{3}{2}$.

B. The line is solid with a $y$-intercept at $(0, -3)$ and slope of $\frac{3}{2}$.

Our findings match option B, which states that the line is solid with a $y$-intercept at $(0, -3)$ and slope of $\frac{3}{2}$.

Final Answer

The final answer is that the attributes of the boundary line of the inequality $-3x - 2y < 6$ are:

• Slope: $\frac{3}{2}$
• $y$-Intercept: $-3$

The boundary line is a solid line with a $y$-intercept at $(0, -3)$ and slope of $\frac{3}{2}$.

Frequently Asked Questions

Q1: What is the boundary line of an inequality?

A1: The boundary line of an inequality is the line that separates the region where the inequality is true from the region where it is false. In other words, it is the line that represents the boundary between the two regions.

Q2: How do I find the attributes of the boundary line of an inequality?

A2: To find the attributes of the boundary line of an inequality, you need to convert the inequality into the slope-intercept form, which is $y = mx + b$. The slope ($m$) and $y$-intercept ($b$) of the boundary line are the attributes you need to find.

Q3: What is the slope of the boundary line?

A3: The slope of the boundary line is the coefficient of $x$ in the slope-intercept form of the inequality. It represents the rate of change of the line.

Q4: What is the $y$-intercept of the boundary line?

A4: The $y$-intercept of the boundary line is the constant term in the slope-intercept form of the inequality. It represents the point where the line intersects the $y$-axis.

Q5: How do I determine if the boundary line is solid or dashed?

A5: The boundary line is solid if the inequality is strict (less than or greater than), and it is dashed if the inequality is non-strict (less than or equal to or greater than or equal to).

Q6: What is the significance of the boundary line in an inequality?

A6: The boundary line is significant because it separates the region where the inequality is true from the region where it is false. It helps to visualize the solution to the inequality.

Q7: Can the boundary line be a vertical line?

A7: Yes, the boundary line can be a vertical line if the inequality is in the form $x = a$, where $a$ is a constant.

Q8: Can the boundary line be a horizontal line?

A8: Yes, the boundary line can be a horizontal line if the inequality is in the form $y = b$, where $b$ is a constant.

Q9: How do I graph the boundary line?

A9: To graph the boundary line, you need to plot the $y$-intercept and then use the slope to find another point on the line. You can then draw a line through the two points.

Q10: What is the relationship between the boundary line and the solution to the inequality?

A10: The boundary line is the boundary between the region where the inequality is true and the region where it is false. The solution to the inequality is the region where the inequality is true.

Conclusion

In conclusion, the attributes of the boundary line of an inequality are the slope and $y$-intercept. The boundary line is solid if the inequality is strict and dashed if the inequality is non-strict. The boundary line is significant because it separates the region where the inequality is true from the region where it is false.