Solve The Inequality: 3 B + 12 \textgreater 27 − 1 2 B 3b + 12 \ \textgreater \ 27 - \frac{1}{2b} 3 B + 12 \textgreater 27 − 2 B 1 ​

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Introduction

In this article, we will focus on solving the given inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$. This involves isolating the variable $b$ and determining the values of $b$ that satisfy the given inequality. We will use algebraic manipulations and properties of inequalities to solve this problem.

Understanding the Inequality

The given inequality is $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$. To begin solving this inequality, we need to simplify it by combining like terms and eliminating any fractions. We can start by subtracting $12$ from both sides of the inequality, which gives us:

$$3b \ \textgreater \ 15 - \frac{1}{2b}$$

Eliminating the Fraction

To eliminate the fraction, we can multiply both sides of the inequality by $2b$. However, we need to be careful when multiplying both sides of an inequality by a variable expression. If the variable expression is negative, the direction of the inequality will be reversed. In this case, we assume that $2b$ is positive, so the direction of the inequality remains the same.

$$6b^2 \ \textgreater \ 30 - 1$$

Simplifying the Inequality

Now, we can simplify the inequality by combining the constant terms on the right-hand side.

$$6b^2 \ \textgreater \ 29$$

Solving for $b$

To solve for $b$, we can take the square root of both sides of the inequality. However, we need to be careful when taking the square root of both sides of an inequality. If the expression inside the square root is negative, the direction of the inequality will be reversed. In this case, we assume that $6b^2$ is positive, so the direction of the inequality remains the same.

$$\sqrt{6b^2} \ \textgreater \ \sqrt{29}$$

Simplifying the Square Root

Now, we can simplify the square root by factoring out the perfect square.

$$\sqrt{6}b \ \textgreater \ \sqrt{29}$$

Solving for $b$

To solve for $b$, we can divide both sides of the inequality by $\sqrt{6}$.

$$b \ \textgreater \ \frac{\sqrt{29}}{\sqrt{6}}$$

Simplifying the Fraction

Now, we can simplify the fraction by rationalizing the denominator.

$$b \ \textgreater \ \frac{\sqrt{29}\sqrt{6}}{\sqrt{6}\sqrt{6}}$$

Simplifying the Square Root

Now, we can simplify the square root by combining the square roots in the denominator.

$$b \ \textgreater \ \frac{\sqrt{174}}{6}$$

Conclusion

In this article, we solved the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$ by isolating the variable $b$ and determining the values of $b$ that satisfy the given inequality. We used algebraic manipulations and properties of inequalities to solve this problem. The solution to the inequality is $b \ \textgreater \ \frac{\sqrt{174}}{6}$.

Final Answer

The final answer is $\boxed{\frac{\sqrt{174}}{6}}$.

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Introduction

In our previous article, we solved the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$ by isolating the variable $b$ and determining the values of $b$ that satisfy the given inequality. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

Q: What is the first step in solving the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$?

A: The first step in solving the inequality is to simplify it by combining like terms and eliminating any fractions. We can start by subtracting $12$ from both sides of the inequality, which gives us $3b \ \textgreater \ 15 - \frac{1}{2b}$.

Q: Why do we need to be careful when multiplying both sides of an inequality by a variable expression?

A: When multiplying both sides of an inequality by a variable expression, we need to be careful because the direction of the inequality may be reversed if the variable expression is negative. In this case, we assume that $2b$ is positive, so the direction of the inequality remains the same.

Q: How do we eliminate the fraction in the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$?

A: To eliminate the fraction, we can multiply both sides of the inequality by $2b$. However, as mentioned earlier, we need to be careful when multiplying both sides of an inequality by a variable expression.

Q: What is the final solution to the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$?

A: The final solution to the inequality is $b \ \textgreater \ \frac{\sqrt{174}}{6}$.

Q: Can you explain why we take the square root of both sides of the inequality?

A: We take the square root of both sides of the inequality because we want to isolate the variable $b$. By taking the square root of both sides, we can simplify the inequality and solve for $b$.

Q: What is the significance of the square root in the inequality?

A: The square root in the inequality represents the positive and negative square roots of the expression inside the square root. In this case, we are only interested in the positive square root because we are solving for $b$.

Q: Can you provide an example of how to use the solution to the inequality in a real-world problem?

A: Suppose we are given a problem where we need to find the values of $b$ that satisfy the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$. Using the solution to the inequality, we can determine that $b \ \textgreater \ \frac{\sqrt{174}}{6}$. This means that any value of $b$ greater than $\frac{\sqrt{174}}{6}$ will satisfy the inequality.

Conclusion

In this article, we provided a Q&A section to help clarify any doubts or questions that readers may have about solving the inequality $3b + 12 \ \textgreater \ 27 - \frac{1}{2b}$. We hope that this Q&A section has been helpful in understanding the solution to the inequality.

Final Answer

The final answer is $\boxed{\frac{\sqrt{174}}{6}}$.