Which Graph Best Represents The Solution Set Of − 4 X ≤ 6 Y − 54 -4x \leq 6y - 54 − 4 X ≤ 6 Y − 54 ?

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Introduction

Graphing linear inequalities is a crucial concept in mathematics, particularly in algebra and geometry. It involves representing the solution set of an inequality on a coordinate plane. In this article, we will explore how to graph the solution set of the inequality $-4x \leq 6y - 54$ and determine which graph best represents it.

Understanding the Inequality

The given inequality is $-4x \leq 6y - 54$. To graph this inequality, we need to first rewrite it in the slope-intercept form, which is $y \geq mx + b$, where $m$ is the slope and $b$ is the y-intercept.

Step 1: Rewrite the Inequality

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

$$-4x \leq 6y - 54$$

Add $4x$ to both sides:

$$0 \leq 6y - 4x - 54$$

Add $4x$ to both sides:

$$4x \leq 6y - 54$$

Add $54$ to both sides:

$$4x + 54 \leq 6y$$

Divide both sides by $6$:

$$\frac{4x + 54}{6} \leq y$$

Simplify the expression:

$$\frac{2x + 27}{3} \leq y$$

Step 2: Graph the Inequality

Now that we have rewritten the inequality in the slope-intercept form, we can graph it on a coordinate plane.

The inequality is $y \geq \frac{2x + 27}{3}$. This means that the solution set is all the points on or above the line $y = \frac{2x + 27}{3}$.

Graphing the Solution Set

To graph the solution set, we need to plot the line $y = \frac{2x + 27}{3}$ and shade the region above it.

Step 1: Plot the Line

To plot the line, we need to find the x-intercept and the y-intercept.

The x-intercept is the point where the line intersects the x-axis. To find the x-intercept, we set $y = 0$ and solve for $x$.

$$0 = \frac{2x + 27}{3}$$

Multiply both sides by $3$:

$$0 = 2x + 27$$

Subtract $27$ from both sides:

$$-27 = 2x$$

Divide both sides by $2$:

$$x = -\frac{27}{2}$$

The x-intercept is $(-\frac{27}{2}, 0)$.

The y-intercept is the point where the line intersects the y-axis. To find the y-intercept, we set $x = 0$ and solve for $y$.

$$y = \frac{2(0) + 27}{3}$$

Simplify the expression:

$$y = \frac{27}{3}$$

$$y = 9$$

The y-intercept is $(0, 9)$.

Step 2: Plot the Points

Now that we have found the x-intercept and the y-intercept, we can plot the points on the coordinate plane.

Plot the point $(-\frac{27}{2}, 0)$ on the x-axis.

Plot the point $(0, 9)$ on the y-axis.

Step 3: Draw the Line

Now that we have plotted the points, we can draw the line $y = \frac{2x + 27}{3}$.

Draw a line through the points $(-\frac{27}{2}, 0)$ and $(0, 9)$.

Step 4: Shade the Region

Now that we have drawn the line, we can shade the region above it.

Shade the region above the line $y = \frac{2x + 27}{3}$.

Conclusion

In this article, we have explored how to graph the solution set of the inequality $-4x \leq 6y - 54$. We have rewritten the inequality in the slope-intercept form, plotted the line, and shaded the region above it. The graph that best represents the solution set is the one with the line $y = \frac{2x + 27}{3}$ and the shaded region above it.

Final Answer

The final answer is the graph with the line $y = \frac{2x + 27}{3}$ and the shaded region above it.

Graphs to Compare

Here are some graphs to compare with the correct graph:

Graph 1

This graph has the line $y = \frac{2x + 27}{3}$, but it does not have the shaded region above it.

Graph 2

This graph has the line $y = \frac{2x + 27}{3}$ and the shaded region above it, but it is not the correct graph because the shaded region is below the line.

Graph 3

This graph has the line $y = \frac{2x + 27}{3}$ and the shaded region above it, but it is not the correct graph because the line is not drawn correctly.

Comparison of Graphs

Here is a comparison of the graphs:

Graph	Line	Shaded Region
Correct Graph	$y = \frac{2x + 27}{3}$	Above the line
Graph 1	$y = \frac{2x + 27}{3}$	None
Graph 2	$y = \frac{2x + 27}{3}$	Below the line
Graph 3	Incorrect line	Above the line

Conclusion

In conclusion, the graph that best represents the solution set of the inequality $-4x \leq 6y - 54$ is the one with the line $y = \frac{2x + 27}{3}$ and the shaded region above it.

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Introduction

In our previous article, we explored how to graph the solution set of the inequality $-4x \leq 6y - 54$. We rewrote the inequality in the slope-intercept form, plotted the line, and shaded the region above it. In this article, we will answer some frequently asked questions about graphing the solution set of this inequality.

Q1: What is the slope of the line $y = \frac{2x + 27}{3}$?

A1: The slope of the line $y = \frac{2x + 27}{3}$ is $\frac{2}{3}$.

Q2: What is the y-intercept of the line $y = \frac{2x + 27}{3}$?

A2: The y-intercept of the line $y = \frac{2x + 27}{3}$ is $9$.

Q3: How do I determine the direction of the inequality?

A3: To determine the direction of the inequality, you need to look at the sign of the coefficient of $x$. If the coefficient is positive, the inequality is pointing upwards. If the coefficient is negative, the inequality is pointing downwards.

Q4: Can I use a graphing calculator to graph the solution set?

A4: Yes, you can use a graphing calculator to graph the solution set. However, make sure to set the calculator to the correct mode and enter the inequality correctly.

Q5: How do I know if the graph is correct?

A5: To check if the graph is correct, you need to make sure that the line is drawn correctly and the shaded region is above the line.

Q6: Can I graph the solution set of a system of inequalities?

A6: Yes, you can graph the solution set of a system of inequalities. However, you need to graph each inequality separately and then find the intersection of the two graphs.

Q7: How do I graph the solution set of an inequality with a fraction?

A7: To graph the solution set of an inequality with a fraction, you need to rewrite the inequality in the slope-intercept form and then graph the line.

Q8: Can I use a graphing software to graph the solution set?

A8: Yes, you can use a graphing software to graph the solution set. However, make sure to set the software to the correct mode and enter the inequality correctly.

Q9: How do I determine the boundary line of the solution set?

A9: The boundary line of the solution set is the line that separates the region above the line from the region below the line.

Q10: Can I graph the solution set of an inequality with a negative slope?

A10: Yes, you can graph the solution set of an inequality with a negative slope. However, you need to make sure to draw the line correctly and shade the region above or below the line.

Conclusion

In conclusion, graphing the solution set of an inequality is a crucial concept in mathematics. By following the steps outlined in this article, you can graph the solution set of the inequality $-4x \leq 6y - 54$ and determine the direction of the inequality. Remember to use a graphing calculator or software to check your work and make sure the graph is correct.

Final Answer

The final answer is the graph with the line $y = \frac{2x + 27}{3}$ and the shaded region above it.

Additional Resources

Here are some additional resources to help you graph the solution set of an inequality:

• Graphing calculators: TI-83, TI-84, TI-Nspire
• Graphing software: GeoGebra, Desmos, Graphing Calculator
• Online resources: Khan Academy, Mathway, Wolfram Alpha

Conclusion

In conclusion, graphing the solution set of an inequality is a crucial concept in mathematics. By following the steps outlined in this article, you can graph the solution set of the inequality $-4x \leq 6y - 54$ and determine the direction of the inequality. Remember to use a graphing calculator or software to check your work and make sure the graph is correct.