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

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In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. Solving inequalities requires a different approach than solving equations, and it's essential to understand the properties of inequalities to solve them correctly. In this article, we will focus on solving the inequality $3\left(\frac{1}{4}\right)^{n+1}\ \textless \ 192$ and provide a step-by-step guide on how to solve it.

Understanding the Inequality

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The given inequality is $3\left(\frac{1}{4}\right)^{n+1}\ \textless \ 192$. To solve this inequality, we need to isolate the variable $n$ and find the values of $n$ that satisfy the inequality.

Step 1: Simplify the Inequality

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The first step in solving the inequality is to simplify it by getting rid of the fraction. We can do this by multiplying both sides of the inequality by 4, which is the denominator of the fraction.

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

$$\Rightarrow 3\left(\frac{1}{4}\right)^{n+1}\ \textless \ 768$$

Step 2: Divide Both Sides by 3

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Next, we can divide both sides of the inequality by 3 to get rid of the coefficient 3.

$$\frac{3\left(\frac{1}{4}\right)^{n+1}}{3}\ \textless \ \frac{768}{3}$$

$$\Rightarrow \left(\frac{1}{4}\right)^{n+1}\ \textless \ 256$$

Step 3: Take the Logarithm of Both Sides

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To solve the inequality, we can take the logarithm of both sides. We can use any base for the logarithm, but it's common to use the natural logarithm (base e) or the logarithm to the base 10.

$$\Rightarrow \ln\left(\left(\frac{1}{4}\right)^{n+1}\right)\ \textless \ \ln(256)$$

Step 4: Use the Property of Logarithms

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Using the property of logarithms that states $\ln(a^b) = b\ln(a)$, we can rewrite the left-hand side of the inequality.

$$\Rightarrow (n+1)\ln\left(\frac{1}{4}\right)\ \textless \ \ln(256)$$

Step 5: Simplify the Inequality

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Now, we can simplify the inequality by dividing both sides by $\ln\left(\frac{1}{4}\right)$.

$$\Rightarrow n+1\ \textless \ \frac{\ln(256)}{\ln\left(\frac{1}{4}\right)}$$

Step 6: Evaluate the Right-Hand Side

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To evaluate the right-hand side of the inequality, we can use a calculator to find the value of $\frac{\ln(256)}{\ln\left(\frac{1}{4}\right)}$.

$$\Rightarrow n+1\ \textless \ \frac{\ln(256)}{\ln\left(\frac{1}{4}\right)} \approx \frac{5.545}{-1.386} \approx -4$$

Step 7: Solve for n

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Finally, we can solve for $n$ by subtracting 1 from both sides of the inequality.

$$\Rightarrow n\ \textless \ -4 - 1$$

$$\Rightarrow n\ \textless \ -5$$

Conclusion

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The solution to the inequality $3\left(\frac{1}{4}\right)^{n+1}\ \textless \ 192$ is $n\ \textless \ -5$. This means that the value of $n$ must be less than -5 to satisfy the inequality.

Answer

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The correct answer is:

• A. $x\ \textgreater \ -6$ is incorrect
• B. $x\ \textgreater \ -4$ is incorrect
• C. $x\ \textless \ 4$ is incorrect
• D. $x\ \textless \ -5$ is correct
  

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In the previous article, we solved the inequality $3\left(\frac{1}{4}\right)^{n+1}\ \textless \ 192$ and found that the solution is $n\ \textless \ -5$. However, there are many more questions and scenarios that can arise when dealing with inequalities. In this article, we will provide a Q&A guide to help you understand and solve inequalities.

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

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A: An inequality is a statement that compares two or more mathematical expressions using a relation such as <, >, ≤, or ≥. An equation, on the other hand, is a statement that states that two or more mathematical expressions are equal.

Q: How do I solve an inequality with a variable in the exponent?

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A: To solve an inequality with a variable in the exponent, you can use the following steps:

1. Simplify the inequality by getting rid of any fractions or coefficients.
2. Take the logarithm of both sides of the inequality.
3. Use the property of logarithms to rewrite the left-hand side of the inequality.
4. Simplify the inequality by dividing both sides by the logarithm of the base.
5. Solve for the variable.

Q: What is the difference between a linear inequality and a quadratic inequality?

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A: A linear inequality is an inequality that can be written in the form $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic inequality?

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A: To solve a quadratic inequality, you can use the following steps:

1. Factor the quadratic expression, if possible.
2. Use the factored form to find the roots of the quadratic expression.
3. Use a number line or a graph to determine the intervals where the quadratic expression is positive or negative.
4. Write the solution to the inequality in interval notation.

Q: What is the difference between a rational inequality and a polynomial inequality?

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A: A rational inequality is an inequality that involves a rational expression, such as $\frac{x}{x-1} < 0$. A polynomial inequality, on the other hand, is an inequality that involves a polynomial expression, such as $x^2 + 2x + 1 > 0$.

Q: How do I solve a rational inequality?

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A: To solve a rational inequality, you can use the following steps:

1. Factor the numerator and denominator, if possible.
2. Use the factored form to find the roots of the rational expression.
3. Use a number line or a graph to determine the intervals where the rational expression is positive or negative.
4. Write the solution to the inequality in interval notation.

Q: What is the difference between a system of linear inequalities and a system of quadratic inequalities?

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A: A system of linear inequalities is a set of linear inequalities that must be satisfied simultaneously. A system of quadratic inequalities, on the other hand, is a set of quadratic inequalities that must be satisfied simultaneously.

Q: How do I solve a system of linear inequalities?

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A: To solve a system of linear inequalities, you can use the following steps:

1. Graph the inequalities on a number line or a coordinate plane.
2. Find the intersection of the two inequalities.
3. Write the solution to the system in interval notation.

Q: How do I solve a system of quadratic inequalities?

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A: To solve a system of quadratic inequalities, you can use the following steps:

1. Factor the quadratic expressions, if possible.
2. Use the factored form to find the roots of the quadratic expressions.
3. Graph the quadratic expressions on a number line or a coordinate plane.
4. Find the intersection of the two quadratic expressions.
5. Write the solution to the system in interval notation.

Conclusion

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Solving inequalities can be a challenging task, but with the right tools and techniques, it can be done. In this article, we provided a Q&A guide to help you understand and solve inequalities. We covered topics such as linear inequalities, quadratic inequalities, rational inequalities, and systems of inequalities. By following the steps outlined in this article, you can solve inequalities with confidence and accuracy.

Final Answer

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The final answer is:

• A. $x\ \textgreater \ -6$ is incorrect
• B. $x\ \textgreater \ -4$ is incorrect
• C. $x\ \textless \ 4$ is incorrect
• D. $x\ \textless \ -5$ is correct