Which Graph Shows The Solution Set Of $\frac{x-9}{7x+2} \leq 0$?

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Introduction

In mathematics, graphing inequalities is a crucial concept that helps us visualize the solution sets of various functions. When dealing with rational inequalities, it's essential to understand how to graph the solution set of the inequality. In this article, we will explore how to graph the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$.

Understanding the Inequality

The given inequality is $\frac{x-9}{7x+2} \leq 0$. To graph the solution set, we need to find the critical points where the expression on the left-hand side changes sign. The critical points occur when the numerator or denominator equals zero.

Finding Critical Points

To find the critical points, we set the numerator and denominator equal to zero and solve for x.

• For the numerator: $x - 9 = 0 \Rightarrow x = 9$
• For the denominator: $7x + 2 = 0 \Rightarrow x = -\frac{2}{7}$

Graphing the Solution Set

To graph the solution set, we need to create a sign chart for the expression $\frac{x-9}{7x+2}$. We will divide the number line into intervals based on the critical points and determine the sign of the expression in each interval.

Sign Chart

Interval	Sign of Numerator	Sign of Denominator	Sign of Expression
$(-\infty, -\frac{2}{7})$	-	-	+
$(-\frac{2}{7}, 9)$	-	+	-
$(9, \infty)$	+	+	+

Analyzing the Sign Chart

Based on the sign chart, we can see that the expression $\frac{x-9}{7x+2}$ is negative in the interval $(-\frac{2}{7}, 9)$. This means that the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$ is the interval $(-\frac{2}{7}, 9]$.

Conclusion

In conclusion, to graph the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$, we need to find the critical points, create a sign chart, and analyze the sign chart to determine the solution set. The solution set is the interval $(-\frac{2}{7}, 9]$.

Graphing the Solution Set on a Number Line

To graph the solution set on a number line, we need to plot the critical points and shade the interval $(-\frac{2}{7}, 9]$.

Number Line Representation

[Insert number line with critical points and shaded interval]

Graphing the Solution Set on a Coordinate Plane

To graph the solution set on a coordinate plane, we need to plot the critical points and shade the region that satisfies the inequality.

Coordinate Plane Representation

[Insert coordinate plane with critical points and shaded region]

Final Answer

The final answer is the interval $(-\frac{2}{7}, 9]$.

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Introduction

Graphing inequalities is a crucial concept in mathematics that helps us visualize the solution sets of various functions. In the previous article, we explored how to graph the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$. In this article, we will answer some frequently asked questions (FAQs) about graphing the solution set of this inequality.

Q: What are the critical points of the inequality $\frac{x-9}{7x+2} \leq 0$?

A: The critical points of the inequality $\frac{x-9}{7x+2} \leq 0$ are $x = 9$ and $x = -\frac{2}{7}$.

Q: How do I create a sign chart for the expression $\frac{x-9}{7x+2}$?

A: To create a sign chart, you need to divide the number line into intervals based on the critical points and determine the sign of the expression in each interval. You can use the following steps:

1. Divide the number line into intervals based on the critical points.
2. Determine the sign of the numerator and denominator in each interval.
3. Determine the sign of the expression in each interval by multiplying the signs of the numerator and denominator.

Q: What does the sign chart tell us about the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$?

A: The sign chart tells us that the expression $\frac{x-9}{7x+2}$ is negative in the interval $(-\frac{2}{7}, 9)$. This means that the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$ is the interval $(-\frac{2}{7}, 9]$.

Q: How do I graph the solution set on a number line?

A: To graph the solution set on a number line, you need to plot the critical points and shade the interval $(-\frac{2}{7}, 9]$.

Q: How do I graph the solution set on a coordinate plane?

A: To graph the solution set on a coordinate plane, you need to plot the critical points and shade the region that satisfies the inequality.

Q: What is the final answer to the inequality $\frac{x-9}{7x+2} \leq 0$?

A: The final answer to the inequality $\frac{x-9}{7x+2} \leq 0$ is the interval $(-\frac{2}{7}, 9]$.

Q: Can I use a graphing calculator to graph the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$?

A: Yes, you can use a graphing calculator to graph the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$. However, it's essential to understand the underlying concepts and be able to graph the solution set by hand.

Q: How do I determine the solution set of a rational inequality?

A: To determine the solution set of a rational inequality, you need to follow these steps:

1. Find the critical points of the inequality.
2. Create a sign chart for the expression.
3. Analyze the sign chart to determine the solution set.

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

A: Some common mistakes to avoid when graphing the solution set of a rational inequality include:

• Failing to find the critical points.
• Creating an incorrect sign chart.
• Failing to analyze the sign chart correctly.
• Graphing the solution set incorrectly on a number line or coordinate plane.

Conclusion

In conclusion, graphing the solution set of a rational inequality requires a thorough understanding of the underlying concepts and a step-by-step approach. By following the steps outlined in this article, you can determine the solution set of the inequality $\frac{x-9}{7x+2} \leq 0$ and graph it correctly on a number line or coordinate plane.