Solve The Inequality:$\[ \left(\frac{1}{36}\right)^{6x-3} \ \textgreater \ 6^{3x-9} \\]Find The Range Of \[$ X \$\] Where The Inequality Holds.

$$

Introduction

In this article, we will solve the given inequality $\left(\frac{1}{36}\right)^{6x-3} \ \textgreater \ 6^{3x-9}$ and find the range of $x$ where the inequality holds. We will use various mathematical techniques, including properties of exponents and logarithms, to simplify the inequality and solve for $x$.

Simplifying the Inequality

To simplify the inequality, we can start by expressing $\frac{1}{36}$ as a power of $6$. We know that $36 = 6^2$, so we can write $\frac{1}{36} = \frac{1}{6^2} = 6^{-2}$.

Using this expression, we can rewrite the inequality as:

$$\left(6^{-2}\right)^{6x-3} \ \textgreater \ 6^{3x-9}$$

Applying Exponent Rules

Now, we can apply the exponent rule $\left(a^m\right)^n = a^{mn}$ to simplify the left-hand side of the inequality:

$$6^{-2(6x-3)} \ \textgreater \ 6^{3x-9}$$

Simplifying the Exponents

Next, we can simplify the exponents on both sides of the inequality:

$$6^{-12x+6} \ \textgreater \ 6^{3x-9}$$

Equating the Exponents

Since the bases are the same, we can equate the exponents and solve for $x$:

$$-12x+6 \ \textgreater \ 3x-9$$

Solving for $x$

Now, we can solve for $x$ by isolating the variable on one side of the inequality:

$$-12x+6 \ \textgreater \ 3x-9$$

$$-12x-3x \ \textgreater \ -9-6$$

$$-15x \ \textgreater \ -15$$

$$x \ \textless \ 1$$

Finding the Range of $x$

The inequality holds when $x$ is less than $1$. Therefore, the range of $x$ where the inequality holds is $(-\infty, 1)$.

Conclusion

In this article, we solved the inequality $\left(\frac{1}{36}\right)^{6x-3} \ \textgreater \ 6^{3x-9}$ and found the range of $x$ where the inequality holds. We used various mathematical techniques, including properties of exponents and logarithms, to simplify the inequality and solve for $x$. The range of $x$ where the inequality holds is $(-\infty, 1)$.

Additional Examples

Here are a few additional examples of inequalities that can be solved using similar techniques:

• $\left(\frac{1}{4}\right)^{2x-1} \ \textgreater \ 4^{x-2}$
• $\left(\frac{1}{9}\right)^{3x-2} \ \textless \ 9^{2x-1}$
• $\left(\frac{1}{16}\right)^{4x-2} \ \textgreater \ 16^{2x-3}$

These examples demonstrate the importance of understanding properties of exponents and logarithms in solving inequalities.

Final Thoughts

Solving inequalities is an essential skill in mathematics, and it requires a deep understanding of various mathematical concepts, including properties of exponents and logarithms. By following the steps outlined in this article, you can solve inequalities and find the range of values where the inequality holds. Remember to always simplify the inequality and use properties of exponents and logarithms to solve for the variable.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Note: The references provided are for general mathematical texts and are not specific to the topic of solving inequalities. However, they provide a good foundation for understanding the mathematical concepts required to solve inequalities.

Introduction

In the previous article, we solved the inequality $\left(\frac{1}{36}\right)^{6x-3} \ \textgreater \ 6^{3x-9}$ and found the range of $x$ where the inequality holds. In this article, we will answer some frequently asked questions about solving inequalities.

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

A: The first step in solving an inequality is to simplify the inequality by expressing the numbers and variables in a more manageable form. This may involve rewriting the inequality in terms of a common base or using properties of exponents and logarithms.

Q: How do I know which direction to go when solving an inequality?

A: When solving an inequality, you need to determine the direction of the inequality. If the inequality is of the form $a^x > b^x$, then you need to determine whether $a$ is greater than $b$ or less than $b$. If $a$ is greater than $b$, then the inequality is of the form $a^x > b^x$, and if $a$ is less than $b$, then the inequality is of the form $a^x < b^x$.

Q: What is the difference between solving an inequality and solving an equation?

A: Solving an inequality and solving an equation are two different processes. Solving an equation involves finding the value of the variable that makes the equation true, while solving an inequality involves finding the range of values of the variable that make the inequality true.

Q: Can I use the same techniques to solve all types of inequalities?

A: No, you cannot use the same techniques to solve all types of inequalities. Different types of inequalities require different techniques. For example, if the inequality is of the form $a^x > b^x$, then you can use properties of exponents and logarithms to solve it, but if the inequality is of the form $a^x < b^x$, then you need to use a different technique.

Q: How do I know when to use logarithms to solve an inequality?

A: You should use logarithms to solve an inequality when the inequality is of the form $a^x > b^x$ or $a^x < b^x$, and the bases of the exponents are different. Logarithms can help you simplify the inequality and solve for the variable.

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

A: Yes, you can use a calculator to solve an inequality, but you need to be careful when using a calculator. Make sure that the calculator is set to the correct mode and that you are using the correct function to solve the inequality.

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

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

• Not simplifying the inequality before solving it
• Not using the correct technique to solve the inequality
• Not checking the solution to make sure it satisfies the original inequality
• Not considering all possible solutions to the inequality

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

A: To check your solution to an inequality, you need to plug the solution back into the original inequality and make sure that it satisfies the inequality. If the solution satisfies the inequality, then it is a valid solution.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has many real-world applications, including:

• Modeling population growth and decline
• Modeling the spread of diseases
• Modeling the behavior of financial markets
• Modeling the behavior of physical systems

Conclusion

Solving inequalities is an essential skill in mathematics, and it requires a deep understanding of various mathematical concepts, including properties of exponents and logarithms. By following the steps outlined in this article, you can solve inequalities and find the range of values where the inequality holds. Remember to always simplify the inequality and use properties of exponents and logarithms to solve for the variable.

Additional Resources

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for the Nonmathematician" by Morris Kline

Note: The references provided are for general mathematical texts and are not specific to the topic of solving inequalities. However, they provide a good foundation for understanding the mathematical concepts required to solve inequalities.