Select The Correct Answer.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

Introduction

In mathematics, inequalities are used to describe relationships between variables. One of the key concepts in understanding inequalities is the boundary line, which separates the region where the inequality is true from the region where it is false. In this article, we will explore the attributes of the boundary line of the inequality $-3x - 2y < 6$.

What is a Boundary Line?

A boundary line is a line that separates the region where an inequality is true from the region where it is false. In the case of the inequality $-3x - 2y < 6$, the boundary line is the line that separates the region where $-3x - 2y < 6$ from the region where $-3x - 2y \geq 6$.

Attributes of the Boundary Line

To determine the attributes of the boundary line, we need to rewrite the inequality in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Rewriting the Inequality

To rewrite the inequality, we need to isolate $y$ on one side of the inequality.

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

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

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

Determining the Slope

The slope of the boundary line is the coefficient of $x$ in the rewritten inequality.

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

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

Determining the $y$-Intercept

The $y$-intercept of the boundary line is the value of $y$ when $x = 0$.

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

When $x = 0$, $y > -3$.

Determining the $x$-Intercept

The $x$-intercept of the boundary line is the value of $x$ when $y = 0$.

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

When $y = 0$, $0 > -\frac{3}{2}x - 3$.

Solving for $x$, we get:

$$-\frac{3}{2}x - 3 > 0$$

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

$$x < -2$$

Conclusion

In conclusion, the attributes of the boundary line of the inequality $-3x - 2y < 6$ are:

• The line is solid.
• The $y$-intercept is at $(0, -3)$.
• The slope is $-\frac{3}{2}$.
• The $x$-intercept is at $(-2, 0)$.

Therefore, the correct answer is:

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

Final Answer

The final answer is:

• A. The line is solid with a $y$-intercept at $(0, -3)$ and slope of $-\frac{3}{2}$.
  Understanding the Attributes of the Boundary Line of an Inequality: Q&A ====================================================================

Introduction

In our previous article, we explored the attributes of the boundary line of the inequality $-3x - 2y < 6$. In this article, we will answer some frequently asked questions about the boundary line of an inequality.

Q: What is the boundary line of an inequality?

A: The boundary line of an inequality is a line that separates the region where the inequality is true from the region where it is false.

Q: How do I determine the attributes of the boundary line of an inequality?

A: To determine the attributes of the boundary line of an inequality, you need to rewrite the inequality in the form $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Q: What is the slope of the boundary line of an inequality?

A: The slope of the boundary line of an inequality is the coefficient of $x$ in the rewritten inequality.

Q: How do I determine the $y$-intercept of the boundary line of an inequality?

A: The $y$-intercept of the boundary line of an inequality is the value of $y$ when $x = 0$.

Q: How do I determine the $x$-intercept of the boundary line of an inequality?

A: The $x$-intercept of the boundary line of an inequality is the value of $x$ when $y = 0$.

Q: What is the difference between a solid and a dashed boundary line?

A: A solid boundary line is used for equalities, while a dashed boundary line is used for inequalities.

Q: Can the boundary line of an inequality be a vertical line?

A: Yes, the boundary line of an inequality can be a vertical line. This occurs when the inequality is in the form $x = a$, where $a$ is a constant.

Q: Can the boundary line of an inequality be a horizontal line?

A: Yes, the boundary line of an inequality can be a horizontal line. This occurs when the inequality is in the form $y = b$, where $b$ is a constant.

Q: How do I graph the boundary line of an inequality?

A: To graph the boundary line of an inequality, you need to plot the points that satisfy the inequality and draw a line through them.

Q: What is the significance of the boundary line of an inequality?

A: The boundary line of an inequality is significant because it separates the region where the inequality is true from the region where it is false.

Conclusion

In conclusion, the boundary line of an inequality is a line that separates the region where the inequality is true from the region where it is false. The attributes of the boundary line of an inequality include the slope, $y$-intercept, and $x$-intercept. Understanding the boundary line of an inequality is essential in solving linear inequalities.

Final Answer

The final answer is:

• The boundary line of an inequality is a line that separates the region where the inequality is true from the region where it is false.
• The attributes of the boundary line of an inequality include the slope, $y$-intercept, and $x$-intercept.
• The boundary line of an inequality can be a vertical or horizontal line.
• The boundary line of an inequality is significant because it separates the region where the inequality is true from the region where it is false.