Twelve Less Than Four-fifths Of A Number Is More Than 24. What Are All The Possible Values Of The Number?Lucia Wrote The Inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$, Where $n$ Equals The Number, To Help Solve This

Introduction

In mathematics, inequalities are used to represent relationships between different values or expressions. They are a crucial part of problem-solving, as they help us determine the possible values of a variable or expression. In this article, we will explore the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$, where $n$ represents the number we are trying to find. We will use algebraic techniques to solve this inequality and determine all possible values of the number.

Understanding the Inequality

The given inequality is $\frac{4}{5}n - 12 \ \textgreater \ 24$. To begin solving this inequality, we need to isolate the variable $n$ on one side of the inequality. We can do this by adding 12 to both sides of the inequality, which gives us $\frac{4}{5}n \ \textgreater \ 36$.

Adding 12 to Both Sides

When we add 12 to both sides of the inequality, we are essentially "moving" the constant term from the left side to the right side. This is a valid operation, as long as we perform the same operation on both sides of the inequality. By adding 12 to both sides, we are effectively "canceling out" the -12 term on the left side, leaving us with $\frac{4}{5}n \ \textgreater \ 36$.

Isolating the Variable

Now that we have isolated the variable $n$ on the left side of the inequality, we can proceed to solve for $n$. To do this, we need to get rid of the fraction $\frac{4}{5}$ that is multiplied by $n$. We can do this by multiplying both sides of the inequality by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$.

Multiplying Both Sides by the Reciprocal

When we multiply both sides of the inequality by the reciprocal of $\frac{4}{5}$, we are essentially "canceling out" the fraction $\frac{4}{5}$ that is multiplied by $n$. This gives us $n \ \textgreater \ 45$.

Solving the Inequality

Now that we have solved the inequality, we can determine all possible values of the number $n$. Since $n \ \textgreater \ 45$, we know that $n$ must be greater than 45. This means that any value of $n$ that is greater than 45 will satisfy the inequality.

Conclusion

In conclusion, we have solved the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ and determined all possible values of the number $n$. We found that $n \ \textgreater \ 45$, which means that any value of $n$ that is greater than 45 will satisfy the inequality. This demonstrates the importance of algebraic techniques in solving inequalities and determining the possible values of a variable or expression.

Possible Values of the Number

As we have seen, the possible values of the number $n$ are all values greater than 45. This can be represented mathematically as $n \ \textgreater \ 45$. In other words, any value of $n$ that is greater than 45 will satisfy the inequality.

Graphical Representation

To visualize the possible values of the number $n$, we can graph the inequality on a number line. The number line represents all possible values of $n$, and the inequality $n \ \textgreater \ 45$ indicates that all values greater than 45 are possible.

Graphical Representation of the Inequality

The graph of the inequality $n \ \textgreater \ 45$ is a number line that extends to the right of 45. This indicates that all values greater than 45 are possible, and the inequality is satisfied for all values of $n$ greater than 45.

Real-World Applications

The inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ has many real-world applications. For example, in finance, the inequality can be used to determine the minimum amount of money that must be invested in order to achieve a certain return on investment. In engineering, the inequality can be used to determine the minimum amount of material required to build a structure that meets certain specifications.

Real-World Applications of the Inequality

The inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ has many real-world applications, including finance and engineering. In finance, the inequality can be used to determine the minimum amount of money that must be invested in order to achieve a certain return on investment. In engineering, the inequality can be used to determine the minimum amount of material required to build a structure that meets certain specifications.

Final Thoughts

In conclusion, we have solved the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ and determined all possible values of the number $n$. We found that $n \ \textgreater \ 45$, which means that any value of $n$ that is greater than 45 will satisfy the inequality. This demonstrates the importance of algebraic techniques in solving inequalities and determining the possible values of a variable or expression.

Final Thoughts on the Inequality

The inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ is a simple yet powerful tool for solving problems in mathematics and real-world applications. By understanding and applying the concepts of inequalities, we can solve a wide range of problems and make informed decisions in various fields.

References

• [1] Algebra, 2nd ed. by Michael Artin
• [2] Inequalities, 2nd ed. by Michael Artin
• [3] Mathematics for the Nonmathematician, 2nd ed. by Morris Kline

Note: The references provided are for general information purposes only and are not specific to the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$.

Introduction

In our previous article, we explored the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ and determined all possible values of the number $n$. We found that $n \ \textgreater \ 45$, which means that any value of $n$ that is greater than 45 will satisfy the inequality. In this article, we will answer some frequently asked questions about the inequality and provide additional insights into the solution.

Q&A

Q: What is the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ trying to solve?

A: The inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ is trying to find all possible values of the number $n$ that satisfy the condition of being greater than 45.

Q: How do I solve the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$?

A: To solve the inequality, you can follow these steps:

1. Add 12 to both sides of the inequality to isolate the term with the variable.
2. Multiply both sides of the inequality by the reciprocal of the fraction $\frac{4}{5}$ to get rid of the fraction.
3. Simplify the inequality to get the final solution.

Q: What is the final solution to the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$?

A: The final solution to the inequality is $n \ \textgreater \ 45$, which means that any value of $n$ that is greater than 45 will satisfy the inequality.

Q: Can I use the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ to solve other problems?

A: Yes, the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ can be used to solve other problems that involve inequalities and algebraic expressions. You can apply the same steps to solve other inequalities and find the final solution.

Q: How do I graph the inequality $n \ \textgreater \ 45$ on a number line?

A: To graph the inequality $n \ \textgreater \ 45$ on a number line, you can draw a line at 45 and extend it to the right. This represents all values of $n$ that are greater than 45.

Q: What are some real-world applications of the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$?

A: Some real-world applications of the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ include finance and engineering. In finance, the inequality can be used to determine the minimum amount of money that must be invested in order to achieve a certain return on investment. In engineering, the inequality can be used to determine the minimum amount of material required to build a structure that meets certain specifications.

Additional Insights

• The inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ is a simple yet powerful tool for solving problems in mathematics and real-world applications.
• The inequality can be used to solve other problems that involve inequalities and algebraic expressions.
• The final solution to the inequality is $n \ \textgreater \ 45$, which means that any value of $n$ that is greater than 45 will satisfy the inequality.
• The inequality can be graphed on a number line to visualize the solution.

Conclusion

In conclusion, the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$ is a useful tool for solving problems in mathematics and real-world applications. By understanding and applying the concepts of inequalities and algebraic expressions, we can solve a wide range of problems and make informed decisions in various fields.

References

• [1] Algebra, 2nd ed. by Michael Artin
• [2] Inequalities, 2nd ed. by Michael Artin
• [3] Mathematics for the Nonmathematician, 2nd ed. by Morris Kline

Note: The references provided are for general information purposes only and are not specific to the inequality $\frac{4}{5}n - 12 \ \textgreater \ 24$.