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

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 set.

Understanding the Rational Expression

The rational expression $\frac{3x+8}{x-4}$ consists of two parts: the numerator and the denominator. The numerator is a linear expression $3x+8$, and the denominator is also a linear expression $x-4$. To solve the inequality, we need to consider the signs of the numerator and the denominator separately.

Finding the Critical Points

To solve the inequality, we need to find the critical points where the expression changes sign. The critical points occur when the numerator or the denominator equals zero. In this case, the critical points are:

• $x = -\frac{8}{3}$, where the numerator equals zero
• $x = 4$, where the denominator equals zero

Analyzing the Intervals

To solve the inequality, we need to analyze the intervals created by the critical points. The intervals are:

• $(-\infty, -\frac{8}{3})$
• $(-\frac{8}{3}, 4)$
• $(4, \infty)$

Testing the Intervals

To determine the sign of the expression in each interval, we can choose a test point from each interval and substitute it into the expression. Let's choose the following test points:

• $x = -2$ from the interval $(-\infty, -\frac{8}{3})$
• $x = 0$ from the interval $(-\frac{8}{3}, 4)$
• $x = 5$ from the interval $(4, \infty)$

Substituting these test points into the expression, we get:

• $\frac{3(-2)+8}{-2-4} = \frac{-6+8}{-6} = \frac{2}{-6} = -\frac{1}{3}$ (negative)
• $\frac{3(0)+8}{0-4} = \frac{8}{-4} = -2$ (negative)
• $\frac{3(5)+8}{5-4} = \frac{15+8}{1} = 23$ (positive)

Determining the Solution Set

Based on the analysis of the intervals, we can determine the solution set of the inequality. The expression is negative in the intervals $(-\infty, -\frac{8}{3})$ and $(-\frac{8}{3}, 4)$, and positive in the interval $(4, \infty)$. Therefore, the solution set of the inequality 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, analyzed the intervals, and tested the intervals to determine the solution set. The solution set of the inequality is $x \leq -\frac{8}{3}$ or $x > 4$. This inequality is a classic example of a rational inequality, and solving it requires a thorough understanding of algebraic techniques.

Final Answer

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

Introduction

In the previous article, we solved the inequality $\frac{3x+8}{x-4} \geq 0$ using various techniques. However, we understand that some readers may still have questions or doubts about the solution. In this article, we will address some of the frequently asked questions (FAQs) on solving this inequality.

Q: What are the critical points of the inequality?

A: The critical points of the inequality are the values of x that make the numerator or the denominator equal to zero. In this case, the critical points are:

• $x = -\frac{8}{3}$, where the numerator equals zero
• $x = 4$, where the denominator equals zero

Q: How do I determine the sign of the expression in each interval?

A: To determine the sign of the expression in each interval, you can choose a test point from each interval and substitute it into the expression. Let's choose the following test points:

• $x = -2$ from the interval $(-\infty, -\frac{8}{3})$
• $x = 0$ from the interval $(-\frac{8}{3}, 4)$
• $x = 5$ from the interval $(4, \infty)$

Substituting these test points into the expression, we get:

• $\frac{3(-2)+8}{-2-4} = \frac{-6+8}{-6} = \frac{2}{-6} = -\frac{1}{3}$ (negative)
• $\frac{3(0)+8}{0-4} = \frac{8}{-4} = -2$ (negative)
• $\frac{3(5)+8}{5-4} = \frac{15+8}{1} = 23$ (positive)

Q: Why is the solution set $x \leq -\frac{8}{3}$ or $x > 4$?

A: The solution set is $x \leq -\frac{8}{3}$ or $x > 4$ because the expression is negative in the intervals $(-\infty, -\frac{8}{3})$ and $(-\frac{8}{3}, 4)$, and positive in the interval $(4, \infty)$. Therefore, the solution set includes all values of x that make the expression non-negative.

Q: Can I use a calculator to solve the inequality?

A: Yes, you can use a calculator to solve the inequality. However, keep in mind that a calculator may not always provide the exact solution. In this case, we used algebraic techniques to find the solution set.

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

A: To graph the solution set on a number line, you can use a number line and mark the critical points $x = -\frac{8}{3}$ and $x = 4$. Then, you can shade the intervals that make the expression non-negative.

Q: Can I use the same techniques to solve other rational inequalities?

A: Yes, you can use the same techniques to solve other rational inequalities. However, keep in mind that the specific techniques may vary depending on the inequality.

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

A: Some common mistakes to avoid when solving rational inequalities include:

• Not finding the critical points
• Not analyzing the intervals correctly
• Not testing the intervals correctly
• Not considering the signs of the numerator and denominator separately

Conclusion

In this article, we addressed some of the frequently asked questions (FAQs) on solving the inequality $\frac{3x+8}{x-4} \geq 0$. We provided detailed explanations and examples to help readers understand the solution. We hope that this article has been helpful in clarifying any doubts or questions you may have had about solving this inequality.

Final Answer

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