Solve The Inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$.

Introduction

In this article, we will focus on solving the given inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$. This involves finding the values of $m$ that satisfy the given inequality. We will use algebraic manipulation and mathematical reasoning to solve this inequality.

Understanding the Inequality

The given inequality is a compound inequality, which means it consists of two separate inequalities joined by the word "and". The first inequality is $29 \ \textless \ \frac{m^2 + 9}{2}$, and the second inequality is $\frac{m^2 + 9}{2} \ \textless \ 45$. To solve this compound inequality, we need to solve each of the individual inequalities separately.

Solving the First Inequality

To solve the first inequality $29 \ \textless \ \frac{m^2 + 9}{2}$, we can start by multiplying both sides of the inequality by 2 to eliminate the fraction. This gives us $58 \ \textless \ m^2 + 9$. Next, we can subtract 9 from both sides of the inequality to isolate the term $m^2$. This gives us $49 \ \textless \ m^2$.

Solving the Second Inequality

To solve the second inequality $\frac{m^2 + 9}{2} \ \textless \ 45$, we can start by multiplying both sides of the inequality by 2 to eliminate the fraction. This gives us $m^2 + 9 \ \textless \ 90$. Next, we can subtract 9 from both sides of the inequality to isolate the term $m^2$. This gives us $m^2 \ \textless \ 81$.

Combining the Inequalities

Now that we have solved both individual inequalities, we can combine them to get the final solution. We have $49 \ \textless \ m^2$ and $m^2 \ \textless \ 81$. To combine these inequalities, we can take the intersection of the two solution sets. This means that we need to find the values of $m$ that satisfy both inequalities simultaneously.

Finding the Solution Set

To find the solution set, we can start by taking the square root of both sides of the inequality $49 \ \textless \ m^2$. This gives us $\sqrt{49} \ \textless \ |m|$, which simplifies to $7 \ \textless \ |m|$. Next, we can take the square root of both sides of the inequality $m^2 \ \textless \ 81$. This gives us $\sqrt{m^2} \ \textless \ \sqrt{81}$, which simplifies to $|m| \ \textless \ 9$.

Analyzing the Absolute Value Inequalities

Now that we have the absolute value inequalities $7 \ \textless \ |m|$ and $|m| \ \textless \ 9$, we can analyze them separately. The first inequality $7 \ \textless \ |m|$ means that the absolute value of $m$ is greater than 7. This can be broken down into two separate inequalities: $m \ \textgreater \ 7$ and $m \ \textless \ -7$. The second inequality $|m| \ \textless \ 9$ means that the absolute value of $m$ is less than 9. This can be broken down into two separate inequalities: $m \ \textless \ 9$ and $m \ \textgreater \ -9$.

Combining the Absolute Value Inequalities

Now that we have analyzed the absolute value inequalities, we can combine them to get the final solution. We have $m \ \textgreater \ 7$ and $m \ \textless \ -7$, and $m \ \textless \ 9$ and $m \ \textgreater \ -9$. To combine these inequalities, we can take the intersection of the two solution sets. This means that we need to find the values of $m$ that satisfy both inequalities simultaneously.

Finding the Final Solution

To find the final solution, we can combine the inequalities $m \ \textgreater \ 7$ and $m \ \textless \ -7$, and $m \ \textless \ 9$ and $m \ \textgreater \ -9$. This gives us the solution set $-9 \ \textless \ m \ \textless \ -7$ and $7 \ \textless \ m \ \textless \ 9$.

Conclusion

In this article, we have solved the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$. We have used algebraic manipulation and mathematical reasoning to find the values of $m$ that satisfy the given inequality. The final solution is $-9 \ \textless \ m \ \textless \ -7$ and $7 \ \textless \ m \ \textless \ 9$.

Final Answer

The final answer is $\boxed{-9 \ \textless \ m \ \textless \ -7 \ \text{and} \ 7 \ \textless \ m \ \textless \ 9}$.

Introduction

In our previous article, we solved the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$. We used algebraic manipulation and mathematical reasoning to find the values of $m$ that satisfy the given inequality. In this article, we will answer some common questions related to solving this inequality.

Q1: What is the first step in solving the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$?

A1: The first step in solving the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$ is to multiply both sides of the inequality by 2 to eliminate the fraction. This gives us $58 \ \textless \ m^2 + 9$.

Q2: How do we isolate the term $m^2$ in the inequality $58 \ \textless \ m^2 + 9$?

A2: To isolate the term $m^2$ in the inequality $58 \ \textless \ m^2 + 9$, we can subtract 9 from both sides of the inequality. This gives us $49 \ \textless \ m^2$.

Q3: What is the next step in solving the inequality $49 \ \textless \ m^2$?

A3: The next step in solving the inequality $49 \ \textless \ m^2$ is to take the square root of both sides of the inequality. This gives us $\sqrt{49} \ \textless \ |m|$, which simplifies to $7 \ \textless \ |m|$.

Q4: How do we analyze the absolute value inequality $7 \ \textless \ |m|$?

A4: To analyze the absolute value inequality $7 \ \textless \ |m|$, we can break it down into two separate inequalities: $m \ \textgreater \ 7$ and $m \ \textless \ -7$.

Q5: What is the final solution to the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$?

A5: The final solution to the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$ is $-9 \ \textless \ m \ \textless \ -7$ and $7 \ \textless \ m \ \textless \ 9$.

Q6: Can you explain why we need to take the intersection of the two solution sets?

A6: Yes, we need to take the intersection of the two solution sets because we are looking for the values of $m$ that satisfy both inequalities simultaneously. This means that we need to find the values of $m$ that are common to both solution sets.

Q7: How do we find the intersection of the two solution sets?

A7: To find the intersection of the two solution sets, we can combine the inequalities $m \ \textgreater \ 7$ and $m \ \textless \ -7$, and $m \ \textless \ 9$ and $m \ \textgreater \ -9$. This gives us the solution set $-9 \ \textless \ m \ \textless \ -7$ and $7 \ \textless \ m \ \textless \ 9$.

Conclusion

In this article, we have answered some common questions related to solving the inequality $29 \ \textless \ \frac{m^2 + 9}{2} \ \textless \ 45$. We have used algebraic manipulation and mathematical reasoning to find the values of $m$ that satisfy the given inequality. The final solution is $-9 \ \textless \ m \ \textless \ -7$ and $7 \ \textless \ m \ \textless \ 9$.

Final Answer

The final answer is $\boxed{-9 \ \textless \ m \ \textless \ -7 \ \text{and} \ 7 \ \textless \ m \ \textless \ 9}$.