Find The Solution Set Of $3 - 4k - 8 \ \textgreater \ 2k - 13$ If $k$ Has A Domain (factors Of 24).

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Introduction

In mathematics, solving inequalities is a crucial concept that helps us understand the relationship between different variables. In this article, we will focus on finding the solution set of the inequality $3 - 4k - 8 \ \textgreater \ 2k - 13$, given that $k$ has a domain of factors of 24. We will break down the problem step by step, using algebraic manipulations and logical reasoning to arrive at the solution.

Understanding the Domain of $k$

Before we proceed with solving the inequality, it's essential to understand the domain of $k$. The domain of $k$ refers to the set of possible values that $k$ can take. In this case, we are given that $k$ has a domain of factors of 24. This means that $k$ can take on any value that is a factor of 24.

Factors of 24

To find the factors of 24, we can start by listing all the numbers that divide 24 without leaving a remainder. The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

Solving the Inequality

Now that we have a clear understanding of the domain of $k$, we can proceed with solving the inequality. The given inequality is $3 - 4k - 8 \ \textgreater \ 2k - 13$. Our goal is to isolate $k$ and find the solution set.

Step 1: Simplify the Inequality

To simplify the inequality, we can start by combining like terms. We can rewrite the inequality as:

$$-4k - 5 \ \textgreater \ 2k - 13$$

Step 2: Add $4k$ to Both Sides

Next, we can add $4k$ to both sides of the inequality to get:

$$-5 \ \textgreater \ 6k - 13$$

Step 3: Add 13 to Both Sides

Now, we can add 13 to both sides of the inequality to get:

$$8 \ \textgreater \ 6k$$

Step 4: Divide Both Sides by 6

Finally, we can divide both sides of the inequality by 6 to get:

$$\frac{4}{3} \ \textgreater \ k$$

Finding the Solution Set

Now that we have solved the inequality, we need to find the solution set. The solution set refers to the set of all possible values of $k$ that satisfy the inequality.

Since $k$ has a domain of factors of 24, we need to find the factors of 24 that are greater than $\frac{4}{3}$. The factors of 24 that satisfy this condition are: 4, 6, 8, 12, and 24.

Conclusion

In conclusion, the solution set of the inequality $3 - 4k - 8 \ \textgreater \ 2k - 13$ if $k$ has a domain of factors of 24 is: 4, 6, 8, 12, and 24. We arrived at this solution by simplifying the inequality, adding and subtracting terms, and dividing both sides by 6. This problem demonstrates the importance of understanding the domain of a variable and using algebraic manipulations to solve inequalities.

Final Answer

The final answer is: $\boxed{4, 6, 8, 12, 24}$

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Introduction

In our previous article, we solved the inequality $3 - 4k - 8 \ \textgreater \ 2k - 13$ given that $k$ has a domain of factors of 24. We found that the solution set is: 4, 6, 8, 12, and 24. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the problem.

Q&A

Q: What is the domain of $k$?

A: The domain of $k$ refers to the set of possible values that $k$ can take. In this case, we are given that $k$ has a domain of factors of 24.

Q: What are the factors of 24?

A: The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

Q: How do we simplify the inequality?

A: To simplify the inequality, we can start by combining like terms. We can rewrite the inequality as:

$$-4k - 5 \ \textgreater \ 2k - 13$$

Q: Why do we add $4k$ to both sides of the inequality?

A: We add $4k$ to both sides of the inequality to get rid of the negative term. This allows us to isolate $k$ and find the solution set.

Q: Why do we divide both sides of the inequality by 6?

A: We divide both sides of the inequality by 6 to isolate $k$. This allows us to find the solution set and determine the values of $k$ that satisfy the inequality.

Q: What is the solution set of the inequality?

A: The solution set of the inequality is: 4, 6, 8, 12, and 24.

Q: Why are these values the solution set?

A: These values are the solution set because they are the factors of 24 that are greater than $\frac{4}{3}$. This is the condition that we derived from the inequality.

Q: What is the importance of understanding the domain of a variable?

A: Understanding the domain of a variable is crucial in solving inequalities. It helps us determine the possible values of the variable and ensures that we are working with a valid set of solutions.

Q: How can we apply this concept to real-world problems?

A: This concept can be applied to real-world problems in various fields, such as economics, finance, and engineering. For example, in economics, we may need to solve inequalities to determine the optimal price of a product or the maximum profit that can be achieved.

Conclusion

In conclusion, the solution set of the inequality $3 - 4k - 8 \ \textgreater \ 2k - 13$ if $k$ has a domain of factors of 24 is: 4, 6, 8, 12, and 24. We hope that this Q&A section has provided additional insights and helped clarify any doubts. If you have any further questions, please don't hesitate to ask.

Final Answer

The final answer is: $\boxed{4, 6, 8, 12, 24}$