Solve: $\frac{3}{x-4}\ \textgreater \ 0$Choose The Correct Range For X:A. $x\ \textless \ 4$ B. $x\ \textgreater \ -4$ C. $x\ \textgreater \ 4$ D. $x\ \textless \ -4$

Feb 27, 2025 by ADMIN 177 views

Solving Inequalities: A Step-by-Step Guide to Solving the Inequality $\frac{3}{x-4} > 0$

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. One of the most common types of inequalities is the rational inequality, which involves a rational expression. In this article, we will focus on solving the rational inequality $\frac{3}{x-4} > 0$ . We will break down the solution into manageable steps and provide a clear explanation of each step.

The given inequality is $\frac{3}{x-4} > 0$ . To solve this inequality, we need to find the values of $x$ that make the expression $\frac{3}{x-4}$ positive. The expression $\frac{3}{x-4}$ is a rational expression, which means it is the ratio of two polynomials.

Step 1: Find the Critical Points

To solve the inequality, we need to find the critical points, which are the values of $x$ that make the expression $\frac{3}{x-4}$ equal to zero or undefined. In this case, the expression is undefined when $x-4=0$ , which gives us $x=4$ . Therefore, $x=4$ is a critical point.

Step 2: Create a Sign Chart

A sign chart is a useful tool for solving inequalities. It helps us to visualize the sign of the expression $\frac{3}{x-4}$ in different intervals of $x$ . To create a sign chart, we need to identify the intervals of $x$ that make the expression positive or negative.

Step 3: Determine the Sign of the Expression

To determine the sign of the expression $\frac{3}{x-4}$ , we need to consider the signs of the numerator and the denominator. The numerator is a constant, which means it is always positive. The denominator is $x-4$ , which is positive when $x>4$ and negative when $x<4$ .

Step 4: Analyze the Sign Chart

Based on the sign chart, we can see that the expression $\frac{3}{x-4}$ is positive when $x>4$ and negative when $x<4$ . Therefore, the solution to the inequality $\frac{3}{x-4} > 0$ is $x>4$ .

In conclusion, solving the inequality $\frac{3}{x-4} > 0$ requires us to find the critical points, create a sign chart, determine the sign of the expression, and analyze the sign chart. By following these steps, we can determine the solution to the inequality, which is $x>4$ .

The correct answer is:

C. $x > 4$

The correct answer is $x>4$ because the expression $\frac{3}{x-4}$ is positive when $x>4$ and negative when $x<4$ . Therefore, the solution to the inequality $\frac{3}{x-4} > 0$ is $x>4$ .

When solving rational inequalities, it is essential to find the critical points and create a sign chart.
The sign chart helps us to visualize the sign of the expression in different intervals of $x$ .
By analyzing the sign chart, we can determine the solution to the inequality.

Solving inequalities is a crucial concept in mathematics that requires a deep understanding of the subject. By following the steps outlined in this article, we can solve rational inequalities with ease. Remember to find the critical points, create a sign chart, determine the sign of the expression, and analyze the sign chart to determine the solution to the inequality.
Solving Inequalities: A Q&A Guide to Rational Inequalities

In our previous article, we discussed how to solve the rational inequality $\frac{3}{x-4} > 0$ . We broke down the solution into manageable steps and provided a clear explanation of each step. In this article, we will continue to explore rational inequalities and provide a Q&A guide to help you better understand the concept.

Q: What is a rational inequality?

A: A rational inequality is an inequality that involves a rational expression, which is the ratio of two polynomials.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the critical points, create a sign chart, determine the sign of the expression, and analyze the sign chart.

Q: What are critical points?

A: Critical points are the values of $x$ that make the expression equal to zero or undefined.

Q: How do I find critical points?

A: To find critical points, you need to set the denominator equal to zero and solve for $x$ .

Q: What is a sign chart?

A: A sign chart is a table that shows the sign of the expression in different intervals of $x$ .

Q: How do I create a sign chart?

A: To create a sign chart, you need to identify the intervals of $x$ that make the expression positive or negative.

Q: What is the significance of the sign chart?

A: The sign chart helps you to visualize the sign of the expression in different intervals of $x$ .

Q: How do I determine the sign of the expression?

A: To determine the sign of the expression, you need to consider the signs of the numerator and the denominator.

Q: What is the solution to the inequality $\frac{3}{x-4} > 0$ ?

A: The solution to the inequality $\frac{3}{x-4} > 0$ is $x>4$ .

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 critical points
Not creating a sign chart
Not determining the sign of the expression
Not analyzing the sign chart

Q: How can I practice solving rational inequalities?

A: You can practice solving rational inequalities by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

In conclusion, solving rational inequalities requires a deep understanding of the concept. By following the steps outlined in this article, you can solve rational inequalities with ease. Remember to find critical points, create a sign chart, determine the sign of the expression, and analyze the sign chart to determine the solution to the inequality.

When solving rational inequalities, it is essential to find critical points and create a sign chart.
The sign chart helps you to visualize the sign of the expression in different intervals of $x$ .
By analyzing the sign chart, you can determine the solution to the inequality.

Solving rational inequalities is a crucial concept in mathematics that requires a deep understanding of the subject. By following the steps outlined in this article and practicing regularly, you can improve your skills and become proficient in solving rational inequalities.