Graph The Solution Set For This Inequality: − 6 X − 3 Y ≤ − 18 -6x - 3y \leq -18 − 6 X − 3 Y ≤ − 18 Step 1: Identify The X X X - And Y Y Y -intercepts Of The Boundary Line.When X = 0 X = 0 X = 0 , Y = □ Y = \, \square Y = □ When Y = 0 Y = 0 Y = 0 , X = □ X = \, \square X = □

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Step 1: Identify the $x$- and $y$-intercepts of the boundary line.

To begin graphing the solution set for the inequality $-6x - 3y \leq -18$, we need to identify the $x$- and $y$-intercepts of the boundary line. The boundary line is the line that represents the equation $-6x - 3y = -18$. To find the $x$-intercept, we set $y = 0$ and solve for $x$. Similarly, to find the $y$-intercept, we set $x = 0$ and solve for $y$.

When $x = 0$, $y = \, \square$

When $x = 0$, we substitute $x = 0$ into the equation $-6x - 3y = -18$ and solve for $y$.

$$-6(0) - 3y = -18$$

$$-3y = -18$$

$$y = \frac{-18}{-3}$$

$$y = 6$$

So, when $x = 0$, the value of $y$ is $6$. This means that the $y$-intercept is $(0, 6)$.

When $y = 0$, $x = \, \square$

When $y = 0$, we substitute $y = 0$ into the equation $-6x - 3y = -18$ and solve for $x$.

$$-6x - 3(0) = -18$$

$$-6x = -18$$

$$x = \frac{-18}{-6}$$

$$x = 3$$

So, when $y = 0$, the value of $x$ is $3$. This means that the $x$-intercept is $(3, 0)$.

Step 2: Graph the Boundary Line

Now that we have identified the $x$- and $y$-intercepts, we can graph the boundary line. The boundary line is a line that passes through the points $(3, 0)$ and $(0, 6)$. We can use these two points to draw the line.

To graph the line, we can use the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept. The slope of the line is $\frac{6 - 0}{0 - 3} = -2$. The $y$-intercept is $6$. Therefore, the equation of the line is $y = -2x + 6$.

Step 3: Shade the Region

The inequality $-6x - 3y \leq -18$ represents a region in the coordinate plane. To shade the region, we need to determine which side of the boundary line the region lies on. Since the inequality is of the form $ax + by \leq c$, where $a$ and $b$ are positive, the region lies on the side of the boundary line that contains the origin.

To determine which side of the boundary line contains the origin, we can substitute the coordinates of the origin $(0, 0)$ into the inequality and check if it is true.

$$-6(0) - 3(0) \leq -18$$

$$0 \leq -18$$

Since this is false, the region lies on the side of the boundary line that does not contain the origin.

Step 4: Graph the Solution Set

The solution set for the inequality $-6x - 3y \leq -18$ is the region that lies on the side of the boundary line that does not contain the origin. We can graph this region by shading the area on the side of the boundary line that does not contain the origin.

The final graph of the solution set for the inequality $-6x - 3y \leq -18$ is a shaded region that lies on the side of the boundary line that does not contain the origin.

Conclusion

In this article, we have graphed the solution set for the inequality $-6x - 3y \leq -18$. We have identified the $x$- and $y$-intercepts of the boundary line, graphed the boundary line, shaded the region, and graphed the solution set. The final graph of the solution set is a shaded region that lies on the side of the boundary line that does not contain the origin.

Key Takeaways

• To graph the solution set for an inequality, we need to identify the $x$- and $y$-intercepts of the boundary line.
• The boundary line is a line that passes through the points $(3, 0)$ and $(0, 6)$.
• The region lies on the side of the boundary line that does not contain the origin.
• The solution set for the inequality $-6x - 3y \leq -18$ is a shaded region that lies on the side of the boundary line that does not contain the origin.

Practice Problems

1. Graph the solution set for the inequality $2x + 4y \leq 12$.
2. Graph the solution set for the inequality $-3x + 2y \leq 6$.
3. Graph the solution set for the inequality $x - 2y \leq 4$.

Solutions

1. The solution set for the inequality $2x + 4y \leq 12$ is a shaded region that lies on the side of the boundary line that does not contain the origin.
2. The solution set for the inequality $-3x + 2y \leq 6$ is a shaded region that lies on the side of the boundary line that does not contain the origin.
3. The solution set for the inequality $x - 2y \leq 4$ is a shaded region that lies on the side of the boundary line that does not contain the origin.
   Graph the Solution Set for the Inequality: $-6x - 3y \leq -18$ - Q&A ====================================================================

Q: What is the first step in graphing the solution set for the inequality $-6x - 3y \leq -18$?

A: The first step in graphing the solution set for the inequality $-6x - 3y \leq -18$ is to identify the $x$- and $y$-intercepts of the boundary line.

Q: How do I find the $x$-intercept of the boundary line?

A: To find the $x$-intercept of the boundary line, we set $y = 0$ and solve for $x$. We substitute $y = 0$ into the equation $-6x - 3y = -18$ and solve for $x$.

Q: How do I find the $y$-intercept of the boundary line?

A: To find the $y$-intercept of the boundary line, we set $x = 0$ and solve for $y$. We substitute $x = 0$ into the equation $-6x - 3y = -18$ and solve for $y$.

Q: What is the equation of the boundary line?

A: The equation of the boundary line is $y = -2x + 6$. We can find this equation by using the slope-intercept form of a linear equation, which is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

Q: Which side of the boundary line does the region lie on?

A: The region lies on the side of the boundary line that does not contain the origin. We can determine this by substituting the coordinates of the origin $(0, 0)$ into the inequality and checking if it is true.

Q: How do I graph the solution set for the inequality $-6x - 3y \leq -18$?

A: To graph the solution set for the inequality $-6x - 3y \leq -18$, we need to graph the boundary line and shade the region on the side of the boundary line that does not contain the origin.

Q: What is the final graph of the solution set for the inequality $-6x - 3y \leq -18$?

A: The final graph of the solution set for the inequality $-6x - 3y \leq -18$ is a shaded region that lies on the side of the boundary line that does not contain the origin.

Q: What are some key takeaways from graphing the solution set for the inequality $-6x - 3y \leq -18$?

A: Some key takeaways from graphing the solution set for the inequality $-6x - 3y \leq -18$ are:

• To graph the solution set for an inequality, we need to identify the $x$- and $y$-intercepts of the boundary line.
• The boundary line is a line that passes through the points $(3, 0)$ and $(0, 6)$.
• The region lies on the side of the boundary line that does not contain the origin.
• The solution set for the inequality $-6x - 3y \leq -18$ is a shaded region that lies on the side of the boundary line that does not contain the origin.

Q: What are some practice problems for graphing the solution set for inequalities?

A: Some practice problems for graphing the solution set for inequalities are:

1. Graph the solution set for the inequality $2x + 4y \leq 12$.
2. Graph the solution set for the inequality $-3x + 2y \leq 6$.
3. Graph the solution set for the inequality $x - 2y \leq 4$.

Q: What are some solutions to the practice problems?

A: Some solutions to the practice problems are:

1. The solution set for the inequality $2x + 4y \leq 12$ is a shaded region that lies on the side of the boundary line that does not contain the origin.
2. The solution set for the inequality $-3x + 2y \leq 6$ is a shaded region that lies on the side of the boundary line that does not contain the origin.
3. The solution set for the inequality $x - 2y \leq 4$ is a shaded region that lies on the side of the boundary line that does not contain the origin.

Conclusion

In this article, we have answered some common questions about graphing the solution set for the inequality $-6x - 3y \leq -18$. We have covered topics such as identifying the $x$- and $y$-intercepts of the boundary line, graphing the boundary line, shading the region, and graphing the solution set. We have also provided some practice problems and solutions for graphing the solution set for inequalities.