Given The Inequality $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$, Solve For $x$.

Introduction

In this article, we will delve into the world of inequalities and learn how to solve them. Specifically, we will focus on solving the inequality $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$. This type of problem requires a combination of algebraic manipulations and logical reasoning. By the end of this article, you will have a solid understanding of how to approach and solve similar inequalities.

Understanding the Inequality

The given inequality is $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$. To begin solving this inequality, we need to simplify the left-hand side by combining like terms and performing any necessary operations.

Simplifying the Left-Hand Side

To simplify the left-hand side, we need to distribute the negative sign to the terms inside the parentheses and then combine like terms.

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the left-hand side of the inequality
lhs = 10*x - (2/5)*(15*x + 10)

# Simplify the left-hand side
simplified_lhs = sp.simplify(lhs)
print(simplified_lhs)

This code will output the simplified left-hand side of the inequality.

Combining Like Terms

After simplifying the left-hand side, we can combine like terms to further simplify the expression.

# Combine like terms
combined_lhs = sp.simplify(simplified_lhs)
print(combined_lhs)

This code will output the combined left-hand side of the inequality.

Solving the Inequality

Now that we have simplified the left-hand side, we can proceed to solve the inequality.

Isolating the Variable

To isolate the variable $x$, we need to perform a series of algebraic manipulations. We will start by adding $\frac{2}{5}(15x + 10)$ to both sides of the inequality.

# Add 2/5(15x + 10) to both sides
new_inequality = combined_lhs + (2/5)*(15*x + 10) > 48 + (2/5)*(15*x + 10)
print(new_inequality)

This code will output the new inequality after adding $\frac{2}{5}(15x + 10)$ to both sides.

Simplifying the Right-Hand Side

To simplify the right-hand side, we need to distribute the $\frac{2}{5}$ to the terms inside the parentheses.

# Simplify the right-hand side
simplified_rhs = 48 + (2/5)*(15*x + 10)
print(simplified_rhs)

This code will output the simplified right-hand side of the inequality.

Combining Like Terms

After simplifying the right-hand side, we can combine like terms to further simplify the expression.

# Combine like terms
combined_rhs = sp.simplify(simplified_rhs)
print(combined_rhs)

This code will output the combined right-hand side of the inequality.

Final Solution

Now that we have isolated the variable $x$, we can proceed to find the final solution.

Solving for $x$

To solve for $x$, we need to isolate $x$ on one side of the inequality.

# Solve for x
final_solution = sp.solve(combined_lhs > combined_rhs, x)
print(final_solution)

This code will output the final solution to the inequality.

Conclusion

In this article, we learned how to solve the inequality $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$. We started by simplifying the left-hand side, combining like terms, and then isolating the variable $x$. Finally, we solved for $x$ and obtained the final solution. By following these steps, you can solve similar inequalities and become proficient in algebraic manipulations.

Additional Resources

For more information on solving inequalities, check out the following resources:

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

Frequently Asked Questions

Q: What is the final solution to the inequality $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$? A: The final solution is $x > 8$.

Q: How do I simplify the left-hand side of the inequality? A: To simplify the left-hand side, you need to distribute the negative sign to the terms inside the parentheses and then combine like terms.

Q: How do I isolate the variable $x$? A: To isolate the variable $x$, you need to perform a series of algebraic manipulations, including adding $\frac{2}{5}(15x + 10)$ to both sides of the inequality.

$$$$$$$$$$$$$$$$

Introduction

In our previous article, we learned how to solve the inequality $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$. We covered the basics of solving inequalities, including simplifying the left-hand side, combining like terms, and isolating the variable $x$. In this article, we will answer some of the most frequently asked questions about solving inequalities.

Q&A

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

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

Q: How do I simplify the left-hand side of the inequality?

A: To simplify the left-hand side, you need to distribute the negative sign to the terms inside the parentheses and then combine like terms. For example, if you have the inequality $10x - (2x + 3) > 5$, you would first distribute the negative sign to get $10x - 2x - 3 > 5$, and then combine like terms to get $8x - 3 > 5$.

Q: How do I isolate the variable $x$?

A: To isolate the variable $x$, you need to perform a series of algebraic manipulations, including adding or subtracting terms from both sides of the inequality. For example, if you have the inequality $2x + 3 > 5$, you would first subtract 3 from both sides to get $2x > 2$, and then divide both sides by 2 to get $x > 1$.

Q: What is the final solution to the inequality $10x - \frac{2}{5}(15x + 10) \ \textgreater \ 48$?

A: The final solution is $x > 8$.

Q: How do I know if an inequality is linear or quadratic?

A: To determine if an inequality is linear or quadratic, you need to look at the highest power of the variable $x$. If the highest power is 1, the inequality is linear. If the highest power is 2, the inequality is quadratic.

Q: Can I use the same methods to solve quadratic inequalities as I do to solve linear inequalities?

A: No, you cannot use the same methods to solve quadratic inequalities as you do to solve linear inequalities. Quadratic inequalities require a different set of techniques, including factoring and using the quadratic formula.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that uses the symbols $>$ or $<$, while a non-strict inequality is an inequality that uses the symbols $\geq$ or $\leq$.

Q: Can I use the same methods to solve non-strict inequalities as I do to solve strict inequalities?

A: No, you cannot use the same methods to solve non-strict inequalities as you do to solve strict inequalities. Non-strict inequalities require a different set of techniques, including using the concept of equality.

Conclusion

In this article, we answered some of the most frequently asked questions about solving inequalities. We covered topics such as the difference between linear and quadratic inequalities, simplifying the left-hand side, isolating the variable $x$, and more. By understanding these concepts, you will be better equipped to solve inequalities and become proficient in algebraic manipulations.

Additional Resources

For more information on solving inequalities, check out the following resources:

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

Frequently Asked Questions: Solving Inequalities (Part 2)

In our next article, we will continue to answer frequently asked questions about solving inequalities. We will cover topics such as solving quadratic inequalities, using the quadratic formula, and more. Stay tuned for more information!