What Is The Solution Of $\frac{3x+8}{x-4} \geq 0$?A. $x \leq -\frac{8}{3}$ Or $x \ \textgreater \ 4$B. $x \ \textless \ -\frac{8}{3}$ Or $x \ \textgreater \ 4$C. $-\frac{8}{3} \leq X \ \textless \

Introduction

In mathematics, solving inequalities is a crucial aspect of algebra and calculus. Inequalities are mathematical statements that compare two expressions, indicating that one is greater than, less than, or equal to the other. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $\frac{3x+8}{x-4} \geq 0$. This inequality involves a rational expression, and we will use various techniques to find the solution.

Understanding the Rational Expression

The given inequality is $\frac{3x+8}{x-4} \geq 0$. This is a rational expression, which is a fraction of two polynomials. The numerator is $3x+8$, and the denominator is $x-4$. To solve this inequality, we need to find the values of $x$ that make the rational expression non-negative.

Finding the Critical Points

To solve the inequality, we need to find the critical points, which are the values of $x$ that make the numerator or denominator equal to zero. The critical points are the values of $x$ that divide the number line into intervals where the inequality is either true or false.

• The numerator $3x+8$ is equal to zero when $x = -\frac{8}{3}$.
• The denominator $x-4$ is equal to zero when $x = 4$.

Creating a Sign Chart

To solve the inequality, we can create a sign chart, which is a table that shows the sign of the rational expression in each interval. The sign chart is created by testing a value from each interval in the rational expression.

Analyzing the Sign Chart

From the sign chart, we can see that the rational expression is positive in the intervals $(-\infty, -\frac{8}{3})$ and $(4, \infty)$. The rational expression is negative in the interval $(-\frac{8}{3}, 4)$.

Writing the Solution

Based on the sign chart, the solution to the inequality $\frac{3x+8}{x-4} \geq 0$ is $x \leq -\frac{8}{3}$ or $x > 4$.

Conclusion

In this article, we solved the inequality $\frac{3x+8}{x-4} \geq 0$ using various techniques. We found the critical points, created a sign chart, and analyzed the sign chart to determine the solution. The solution to the inequality is $x \leq -\frac{8}{3}$ or $x > 4$.

Frequently Asked Questions

• What is the solution to the inequality $\frac{3x+8}{x-4} \geq 0$?
  • The solution to the inequality is $x \leq -\frac{8}{3}$ or $x > 4$.
• How do I find the critical points of a rational expression?
  • To find the critical points, set the numerator and denominator equal to zero and solve for $x$.
• What is a sign chart, and how do I create one?
  • A sign chart is a table that shows the sign of the rational expression in each interval. To create a sign chart, test a value from each interval in the rational expression.

Final Answer

The final answer is $x \leq -\frac{8}{3}$ or $x > 4$.

Introduction

Solving inequalities is a crucial aspect of mathematics, and it can be a challenging task for many students. In our previous article, we solved the inequality $\frac{3x+8}{x-4} \geq 0$ using various techniques. In this article, we will answer some frequently asked questions about solving inequalities.

Q&A

Q1: What is the solution to the inequality $\frac{3x+8}{x-4} \geq 0$?

A1: The solution to the inequality is $x \leq -\frac{8}{3}$ or $x > 4$.

Q2: How do I find the critical points of a rational expression?

A2: To find the critical points, set the numerator and denominator equal to zero and solve for $x$. For example, in the inequality $\frac{3x+8}{x-4} \geq 0$, the critical points are $x = -\frac{8}{3}$ and $x = 4$.

Q3: What is a sign chart, and how do I create one?

A3: A sign chart is a table that shows the sign of the rational expression in each interval. To create a sign chart, test a value from each interval in the rational expression. For example, in the inequality $\frac{3x+8}{x-4} \geq 0$, the sign chart is:

Q4: How do I determine the solution to an inequality using a sign chart?

A4: To determine the solution to an inequality using a sign chart, look for the intervals where the rational expression is positive. In the inequality $\frac{3x+8}{x-4} \geq 0$, the rational expression is positive in the intervals $(-\infty, -\frac{8}{3})$ and $(4, \infty)$. Therefore, the solution to the inequality is $x \leq -\frac{8}{3}$ or $x > 4$.

Q5: What are some common mistakes to avoid when solving inequalities?

A5: Some common mistakes to avoid when solving inequalities include:

• Not finding all the critical points
• Not creating a sign chart
• Not analyzing the sign chart correctly
• Not writing the solution in interval notation

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to find all the critical points, create a sign chart, and analyze the sign chart correctly to determine the solution to an inequality. If you have any more questions or need further clarification, feel free to ask.

Frequently Asked Questions (FAQs)

• What is the solution to the inequality $\frac{3x+8}{x-4} \geq 0$?
  • The solution to the inequality is $x \leq -\frac{8}{3}$ or $x > 4$.
• How do I find the critical points of a rational expression?
  • To find the critical points, set the numerator and denominator equal to zero and solve for $x$.
• What is a sign chart, and how do I create one?
  • A sign chart is a table that shows the sign of the rational expression in each interval. To create a sign chart, test a value from each interval in the rational expression.
• How do I determine the solution to an inequality using a sign chart?
  • To determine the solution to an inequality using a sign chart, look for the intervals where the rational expression is positive.
• What are some common mistakes to avoid when solving inequalities?
  • Some common mistakes to avoid when solving inequalities include not finding all the critical points, not creating a sign chart, not analyzing the sign chart correctly, and not writing the solution in interval notation.

Final Answer

The final answer is $x \leq -\frac{8}{3}$ or $x > 4$.