Select The Correct Answer.What Is The Solution To This Inequality? 3 ( 1 4 ) X + 1 \textless 192 3\left(\frac{1}{4}\right)^{x+1}\ \textless \ 192 3 ( 4 1 ​ ) X + 1 \textless 192 A. X \textgreater − 4 X \ \textgreater \ -4 X \textgreater − 4 B. X \textgreater − 6 X \ \textgreater \ -6 X \textgreater − 6 C. X \textless 6 X \ \textless \ 6 X \textless 6 D. $x \

Introduction

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving the inequality $3\left(\frac{1}{4}\right)^{x+1}\ \textless \ 192$ and determine the correct solution.

Understanding the Inequality

The given inequality is $3\left(\frac{1}{4}\right)^{x+1}\ \textless \ 192$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to simplify the left-hand side of the inequality.

Simplifying the Left-Hand Side

We can simplify the left-hand side of the inequality by using the properties of exponents. Specifically, we can rewrite $\left(\frac{1}{4}\right)^{x+1}$ as $\frac{1}{4^{x+1}}$. This gives us:

$$3\left(\frac{1}{4}\right)^{x+1}\ \textless \ 192$$

$$3\left(\frac{1}{4^{x+1}}\right)\ \textless \ 192$$

Using Properties of Exponents

We can further simplify the left-hand side of the inequality by using the property of exponents that states $a^{m}a^{n}=a^{m+n}$. In this case, we can rewrite $4^{x+1}$ as $4^{x}\cdot4^{1}$. This gives us:

$$3\left(\frac{1}{4^{x+1}}\right)\ \textless \ 192$$

$$3\left(\frac{1}{4^{x}\cdot4^{1}}\right)\ \textless \ 192$$

Simplifying the Fraction

We can simplify the fraction on the left-hand side of the inequality by canceling out the common factor of $4^{x}$. This gives us:

$$3\left(\frac{1}{4^{x}\cdot4^{1}}\right)\ \textless \ 192$$

$$\frac{3}{4^{x+1}}\ \textless \ 192$$

Multiplying Both Sides

To get rid of the fraction on the left-hand side of the inequality, we can multiply both sides by $4^{x+1}$. This gives us:

$$\frac{3}{4^{x+1}}\ \textless \ 192$$

$$3\ \textless \ 192\cdot4^{x+1}$$

Simplifying the Right-Hand Side

We can simplify the right-hand side of the inequality by using the property of exponents that states $a^{m}a^{n}=a^{m+n}$. In this case, we can rewrite $4^{x+1}$ as $4^{x}\cdot4^{1}$. This gives us:

$$3\ \textless \ 192\cdot4^{x+1}$$

$$3\ \textless \ 192\cdot4^{x}\cdot4^{1}$$

Simplifying the Right-Hand Side (continued)

We can simplify the right-hand side of the inequality by canceling out the common factor of $4^{x}$. This gives us:

$$3\ \textless \ 192\cdot4^{x}\cdot4^{1}$$

$$3\ \textless \ 192\cdot4^{x}\cdot4$$

Simplifying the Right-Hand Side (continued)

We can simplify the right-hand side of the inequality by multiplying $192$ and $4$. This gives us:

$$3\ \textless \ 192\cdot4^{x}\cdot4$$

$$3\ \textless \ 768\cdot4^{x}$$

Dividing Both Sides

To isolate the term with the exponent, we can divide both sides of the inequality by $768$. This gives us:

$$3\ \textless \ 768\cdot4^{x}$$

$$\frac{3}{768}\ \textless \ 4^{x}$$

Simplifying the Left-Hand Side

We can simplify the left-hand side of the inequality by dividing $3$ by $768$. This gives us:

$$\frac{3}{768}\ \textless \ 4^{x}$$

$$\frac{1}{256}\ \textless \ 4^{x}$$

Using Properties of Exponents

We can rewrite the left-hand side of the inequality as a power of $4$. Specifically, we can rewrite $\frac{1}{256}$ as $4^{-8}$. This gives us:

$$\frac{1}{256}\ \textless \ 4^{x}$$

$$4^{-8}\ \textless \ 4^{x}$$

Comparing Exponents

Since the bases are the same, we can compare the exponents. This gives us:

$$-8\ \textless \ x$$

Conclusion

In conclusion, the solution to the inequality $3\left(\frac{1}{4}\right)^{x+1}\ \textless \ 192$ is $x\ \textgreater \ -6$. This means that any value of $x$ greater than $-6$ will satisfy the inequality.

Answer

The correct answer is:

• B. $x \ \textgreater \ -6$

Discussion

This problem requires the use of properties of exponents and inequalities. The key concept is to isolate the term with the exponent and then compare the exponents. This problem is a good example of how to solve inequalities involving exponents.

Additional Examples

Here are a few additional examples of inequalities involving exponents:

• $2^{x+2}\ \textless \ 32$
• $3^{x-1}\ \textgreater \ 27$
• $4^{x+3}\ \textless \ 64$

These examples require the same steps as the original problem: simplifying the left-hand side, using properties of exponents, and comparing exponents.

Conclusion

Q&A: Solving Inequalities Involving Exponents

Q: What is the first step in solving an inequality involving exponents? A: The first step in solving an inequality involving exponents is to simplify the left-hand side of the inequality.

Q: How do I simplify the left-hand side of the inequality? A: To simplify the left-hand side of the inequality, you can use the properties of exponents. Specifically, you can rewrite the expression as a power of the base.

Q: What are some common properties of exponents that I can use to simplify the left-hand side of the inequality? A: Some common properties of exponents that you can use to simplify the left-hand side of the inequality include:

• Product of Powers: $a^{m}a^{n}=a^{m+n}$
• Power of a Power: $(a^{m})^{n}=a^{mn}$
• Power of a Product: $(ab)^{m}=a^{m}b^{m}$

Q: How do I use the properties of exponents to simplify the left-hand side of the inequality? A: To use the properties of exponents to simplify the left-hand side of the inequality, you can follow these steps:

1. Identify the base and the exponent in the expression.
2. Use the properties of exponents to rewrite the expression as a power of the base.
3. Simplify the expression by combining like terms.

Q: What is the next step in solving the inequality? A: The next step in solving the inequality is to isolate the term with the exponent.

Q: How do I isolate the term with the exponent? A: To isolate the term with the exponent, you can use the following steps:

1. Divide both sides of the inequality by the coefficient of the term with the exponent.
2. Simplify the expression by canceling out any common factors.

Q: What is the final step in solving the inequality? A: The final step in solving the inequality is to compare the exponents.

Q: How do I compare the exponents? A: To compare the exponents, you can use the following steps:

1. Identify the bases of the two expressions.
2. Compare the exponents of the two expressions.
3. Determine the relationship between the two expressions based on the comparison of the exponents.

Q: What are some common mistakes to avoid when solving inequalities involving exponents? A: Some common mistakes to avoid when solving inequalities involving exponents include:

• Not simplifying the left-hand side of the inequality: Make sure to simplify the left-hand side of the inequality before isolating the term with the exponent.
• Not isolating the term with the exponent: Make sure to isolate the term with the exponent before comparing the exponents.
• Not comparing the exponents correctly: Make sure to compare the exponents correctly by identifying the bases and the exponents of the two expressions.

Q: How can I practice solving inequalities involving exponents? A: You can practice solving inequalities involving exponents by working through examples and exercises. Some resources for practicing solving inequalities involving exponents include:

• Textbooks and workbooks: Many textbooks and workbooks include examples and exercises on solving inequalities involving exponents.
• Online resources: There are many online resources available for practicing solving inequalities involving exponents, including video tutorials and practice problems.
• Practice tests: You can also practice solving inequalities involving exponents by taking practice tests and quizzes.

Conclusion

In conclusion, solving inequalities involving exponents requires the use of properties of exponents and inequalities. The key concept is to isolate the term with the exponent and then compare the exponents. By following the steps outlined in this article, you can solve inequalities involving exponents and improve your math skills.