Solve The Inequality And Graph The Solution Set On A Real Number Line. Express The Solution Set In Interval Notation.${\frac{6}{x+4} \ \textgreater \ \frac{6}{x-1}}$The Solution Set Is { \square$}$.(Type Your Answer In Interval

Introduction

In this article, we will explore the process of solving inequalities and graphing the solution set on a real number line. We will focus on the given inequality $\frac{6}{x+4} \ \textgreater \ \frac{6}{x-1}$ and express the solution set in interval notation.

Understanding the Inequality

The given inequality is $\frac{6}{x+4} \ \textgreater \ \frac{6}{x-1}$. To solve this inequality, we need to first understand the concept of inequalities and how to manipulate them. An inequality is a statement that compares two expressions and indicates whether one is greater than, less than, or equal to the other.

Solving the Inequality

To solve the inequality, we can start by eliminating the fractions. We can do this by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators, which is $(x+4)(x-1)$.

\frac{6}{x+4} \  \textgreater \  \frac{6}{x-1}

Multiplying both sides by $(x+4)(x-1)$, we get:

6(x-1) \  \textgreater \  6(x+4)

Expanding the left-hand side, we get:

6x-6 \  \textgreater \  6x+24

Subtracting $6x$ from both sides, we get:

-6 \  \textgreater \  24

This is a contradiction, as $-6$ is not greater than $24$. Therefore, the inequality has no solution.

Graphing the Solution Set

Since the inequality has no solution, the solution set is empty. However, we can still graph the solution set on a real number line.

\text{Solution set: } \emptyset

The solution set is represented by a blank or empty set, indicating that there are no values of $x$ that satisfy the inequality.

Conclusion

In this article, we solved the inequality $\frac{6}{x+4} \ \textgreater \ \frac{6}{x-1}$ and expressed the solution set in interval notation. We found that the inequality has no solution, and the solution set is empty. We also graphed the solution set on a real number line.

Key Takeaways

• To solve an inequality, we need to first understand the concept of inequalities and how to manipulate them.
• We can eliminate fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.
• If the inequality has no solution, the solution set is empty.
• We can graph the solution set on a real number line using interval notation.

Final Answer

Introduction

In our previous article, we explored the process of solving inequalities and graphing the solution set on a real number line. We focused on the given inequality $\frac{6}{x+4} \ \textgreater \ \frac{6}{x-1}$ and expressed the solution set in interval notation. In this article, we will answer some frequently asked questions (FAQs) related to solving inequalities and graphing solution sets on a real number line.

Q&A

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to understand the concept of inequalities and how to manipulate them. We need to identify the type of inequality (e.g., linear, quadratic, rational) and determine the appropriate method to solve it.

Q: How do I eliminate fractions in an inequality?

A: To eliminate fractions in an inequality, we can multiply both sides of the inequality by the least common multiple (LCM) of the denominators. This will help us to simplify the inequality and make it easier to solve.

Q: What is the difference between a solution set and an interval?

A: A solution set is the set of all values that satisfy the inequality, while an interval is a way to represent the solution set on a real number line. Intervals are used to indicate the range of values that satisfy the inequality.

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

A: To graph the solution set on a real number line, we need to identify the interval that represents the solution set. We can use a number line to plot the interval and indicate the solution set.

Q: What if the inequality has no solution?

A: If the inequality has no solution, the solution set is empty. We can represent the solution set as a blank or empty set, indicating that there are no values of $x$ that satisfy the inequality.

Q: Can I use a calculator to solve inequalities?

A: Yes, you can use a calculator to solve inequalities. However, it's essential to understand the concept of inequalities and how to manipulate them before using a calculator.

Q: How do I check my solution to an inequality?

A: To check your solution to an inequality, you can substitute the solution into the original inequality and verify that it is true. This will help you to ensure that your solution is correct.

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

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

• Not understanding the concept of inequalities and how to manipulate them
• Not identifying the type of inequality (e.g., linear, quadratic, rational)
• Not eliminating fractions in the inequality
• Not checking the solution to the inequality

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to solving inequalities and graphing solution sets on a real number line. We covered topics such as eliminating fractions, graphing solution sets, and checking solutions. By understanding these concepts and avoiding common mistakes, you can become more confident in solving inequalities and graphing solution sets on a real number line.

Key Takeaways

• To solve an inequality, you need to understand the concept of inequalities and how to manipulate them.
• Eliminating fractions is an essential step in solving inequalities.
• Graphing the solution set on a real number line is a way to represent the solution set.
• Checking the solution to an inequality is crucial to ensure that it is correct.
• Avoiding common mistakes such as not understanding the concept of inequalities and not eliminating fractions can help you to become more confident in solving inequalities and graphing solution sets on a real number line.

Final Answer

The final answer is $\boxed{\emptyset}$.