Solve The Inequality: X + 9 14 \textgreater 1 \frac{x+9}{14}\ \textgreater \ 1 14 X + 9 ​ \textgreater 1

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Introduction

In this article, we will delve into the world of inequalities and learn how to solve a specific inequality of the form $\frac{x+9}{14} > 1$. Inequalities are a fundamental concept in mathematics, and solving them requires a deep understanding of algebraic manipulations and mathematical reasoning. In this discussion, we will break down the steps involved in solving the given inequality and provide a clear explanation of each step.

Understanding the Inequality

The given inequality is $\frac{x+9}{14} > 1$. To solve this inequality, we need to isolate the variable $x$ and determine the values of $x$ that satisfy the inequality. The first step is to understand the inequality and identify the key elements involved.

Key Elements of the Inequality

• The inequality is in the form of a fraction, where the numerator is $x+9$ and the denominator is $14$.
• The inequality is greater than $1$, which means that the value of the fraction must be greater than $1$.
• The variable $x$ is the unknown quantity that we need to solve for.

Step 1: Multiply Both Sides by 14

To solve the inequality, we need to isolate the variable $x$. The first step is to multiply both sides of the inequality by $14$, which is the denominator of the fraction. This will eliminate the fraction and allow us to work with a simpler inequality.

# Multiply both sides by 14
x_plus_9 = 14 * 1

Step 2: Simplify the Inequality

After multiplying both sides by $14$, we get the inequality $x+9 > 14$. The next step is to simplify the inequality by subtracting $9$ from both sides.

# Subtract 9 from both sides
x = 14 - 9

Step 3: Solve for x

Now that we have simplified the inequality, we can solve for $x$. The inequality $x > 5$ means that the value of $x$ must be greater than $5$.

# Solve for x
x = 5

Conclusion

In this article, we solved the inequality $\frac{x+9}{14} > 1$ by following a series of steps. We multiplied both sides by $14$, simplified the inequality, and solved for $x$. The final solution is $x > 5$, which means that the value of $x$ must be greater than $5$.

Graphical Representation

To visualize the solution, we can graph the inequality on a number line. The number line represents the possible values of $x$, and the inequality $x > 5$ means that the value of $x$ must be greater than $5$.

# Graphical representation
import matplotlib.pyplot as plt
x = [5, 10]
y = [0, 0]
plt.plot(x, y, 'ro-')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Inequality')
plt.show()

Final Answer

The final answer is $\boxed{x > 5}$.

Related Topics

• Solving linear inequalities
• Graphing linear inequalities
• Algebraic manipulations

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Note: The references provided are for illustrative purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

In our previous article, we solved the inequality $\frac{x+9}{14} > 1$ by following a series of steps. In this article, we will provide a Q&A section to address any questions or concerns that readers may have.

Q&A

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

A: The first step in solving the inequality is to multiply both sides by $14$, which is the denominator of the fraction.

Q: Why do we multiply both sides by $14$?

A: We multiply both sides by $14$ to eliminate the fraction and simplify the inequality.

Q: What is the next step after multiplying both sides by $14$?

A: After multiplying both sides by $14$, we simplify the inequality by subtracting $9$ from both sides.

Q: What is the final solution to the inequality?

A: The final solution to the inequality is $x > 5$, which means that the value of $x$ must be greater than $5$.

Q: How do we graph the inequality on a number line?

A: To graph the inequality on a number line, we represent the possible values of $x$ and shade the region to the right of $5$.

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

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

• Not multiplying both sides by the same value
• Not simplifying the inequality correctly
• Not considering the direction of the inequality

Q: How do we determine the direction of the inequality?

A: The direction of the inequality is determined by the sign of the coefficient of the variable. In this case, the coefficient of $x$ is positive, so the inequality is greater than.

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

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

• Modeling population growth
• Determining the maximum or minimum value of a function
• Solving optimization problems

Conclusion

In this article, we provided a Q&A section to address any questions or concerns that readers may have about solving the inequality $\frac{x+9}{14} > 1$. We hope that this article has been helpful in clarifying any doubts and providing a better understanding of the topic.

Related Topics

• Solving linear inequalities
• Graphing linear inequalities
• Algebraic manipulations

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "College Algebra" by James Stewart
• [3] "Linear Algebra and Its Applications" by Gilbert Strang

Additional Resources

• Khan Academy: Solving Linear Inequalities
• Mathway: Solving Linear Inequalities
• Wolfram Alpha: Solving Linear Inequalities

Note: The references and additional resources provided are for illustrative purposes only and are not intended to be a comprehensive list of resources on the topic.