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- And Y-intercepts Of The Boundary Line.- When X = 0 X = 0 X = 0 , Y = − 3 Y = -3 Y = − 3 .- When Y = 0 Y = 0 Y = 0 , X = 6 X = 6 X = 6 .

Introduction

In this article, we will explore the process of graphing the solution set for a linear inequality. A linear inequality is an inequality that can be written in the form of $ax + by \leq c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. The solution set for a linear inequality is the set of all points that satisfy the inequality. In this case, we will be graphing the solution set for the inequality $-6x - 3y \leq -18$.

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

To graph the solution set for the inequality, we first need to identify the x- and y-intercepts of the boundary line. The boundary line is the line that divides the solution set into two regions: the region that satisfies the inequality and the region that does not satisfy the inequality.

• When $x = 0$: To find the y-intercept, we substitute $x = 0$ into the inequality and solve for $y$. This gives us $-3y \leq -18$, which simplifies to $y \geq 6$. Therefore, when $x = 0$, $y = -3$.
• When $y = 0$: To find the x-intercept, we substitute $y = 0$ into the inequality and solve for $x$. This gives us $-6x \leq -18$, which simplifies to $x \geq 6$. Therefore, when $y = 0$, $x = 6$.

Step 2: Plot the x- and y-intercepts

Now that we have identified the x- and y-intercepts, we can plot them on a coordinate plane. The x-intercept is the point $(6, 0)$, and the y-intercept is the point $(0, -3)$.

Step 3: Plot the boundary line

The boundary line is a line that passes through the x- and y-intercepts. To plot the boundary 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.

In this case, the slope of the boundary line is $m = \frac{-3}{-6} = \frac{1}{2}$, and the y-intercept is $b = -3$. Therefore, the equation of the boundary line is $y = \frac{1}{2}x - 3$.

Step 4: Shade the solution set

Now that we have plotted the boundary line, we can shade the solution set. The solution set is the region that satisfies the inequality. In this case, the solution set is the region that is on or below the boundary line.

Step 5: Write the solution set in interval notation

Finally, we can write the solution set in interval notation. The solution set is the set of all points that satisfy the inequality. In this case, the solution set is the set of all points that are on or below the boundary line.

The solution set can be written in interval notation as $x \geq 6$ and $y \geq -3$. This can also be written as $x \in [6, \infty)$ and $y \in [-3, \infty)$.

Conclusion

In this article, we have graphed the solution set for the linear inequality $-6x - 3y \leq -18$. We identified the x- and y-intercepts of the boundary line, plotted the boundary line, shaded the solution set, and wrote the solution set in interval notation. The solution set is the region that satisfies the inequality and can be written in interval notation as $x \geq 6$ and $y \geq -3$.

Graphing the Solution Set for a Linear Inequality: Key Takeaways

• Identify the x- and y-intercepts of the boundary line.
• Plot the x- and y-intercepts on a coordinate plane.
• Plot the boundary line using the slope-intercept form of a linear equation.
• Shade the solution set.
• Write the solution set in interval notation.

Introduction

In our previous article, we explored the process of graphing the solution set for a linear inequality. A linear inequality is an inequality that can be written in the form of $ax + by \leq c$, where $a$, $b$, and $c$ are constants, and $x$ and $y$ are variables. The solution set for a linear inequality is the set of all points that satisfy the inequality. In this article, we will answer some frequently asked questions about graphing the solution set for a linear inequality.

Q: What is the boundary line in a linear inequality?

A: The boundary line is the line that divides the solution set into two regions: the region that satisfies the inequality and the region that does not satisfy the inequality. The boundary line is a line that passes through the x- and y-intercepts of the inequality.

Q: How do I find the x- and y-intercepts of the boundary line?

A: To find the x-intercept, substitute $y = 0$ into the inequality and solve for $x$. To find the y-intercept, substitute $x = 0$ into the inequality and solve for $y$.

Q: What is the slope of the boundary line?

A: The slope of the boundary line is the coefficient of $x$ in the inequality. For example, in the inequality $-6x - 3y \leq -18$, the slope is $-6$.

Q: How do I plot the boundary line?

A: To plot the boundary line, 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. Substitute the slope and y-intercept into the equation and plot the line on a coordinate plane.

Q: What is the solution set for a linear inequality?

A: The solution set is the region that satisfies the inequality. In other words, it is the set of all points that are on or below the boundary line.

Q: How do I shade the solution set?

A: To shade the solution set, draw a line on the coordinate plane that represents the boundary line. Then, shade the region that is on or below the boundary line.

Q: Can I write the solution set in interval notation?

A: Yes, you can write the solution set in interval notation. The solution set can be written as $x \geq 6$ and $y \geq -3$. This can also be written as $x \in [6, \infty)$ and $y \in [-3, \infty)$.

Q: What are some common mistakes to avoid when graphing the solution set for a linear inequality?

A: Some common mistakes to avoid when graphing the solution set for a linear inequality include:

• Not identifying the x- and y-intercepts of the boundary line.
• Not plotting the boundary line correctly.
• Not shading the solution set correctly.
• Not writing the solution set in interval notation.

Conclusion

In this article, we have answered some frequently asked questions about graphing the solution set for a linear inequality. We have discussed the boundary line, x- and y-intercepts, slope, plotting the boundary line, shading the solution set, and writing the solution set in interval notation. By following these steps and avoiding common mistakes, you can graph the solution set for any linear inequality.

Graphing the Solution Set for a Linear Inequality: Key Takeaways

• Identify the x- and y-intercepts of the boundary line.
• Plot the x- and y-intercepts on a coordinate plane.
• Plot the boundary line using the slope-intercept form of a linear equation.
• Shade the solution set.
• Write the solution set in interval notation.
• Avoid common mistakes when graphing the solution set for a linear inequality.