Solve The Inequality: ${ 3 - \frac{x}{3} \ \textgreater \ -4 }$

Introduction

Inequalities are mathematical expressions that compare two values, often using greater than, less than, or equal to symbols. Solving inequalities involves isolating the variable on one side of the inequality sign, while maintaining the direction of the inequality. In this article, we will focus on solving the inequality ${ 3 - \frac{x}{3} \ \textgreater \ -4 }$, which involves algebraic manipulation and understanding of inequality properties.

Understanding the Inequality

The given inequality is ${ 3 - \frac{x}{3} \ \textgreater \ -4 }$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to add $\frac{x}{3}$ to both sides of the inequality, which will eliminate the fraction and make it easier to work with.

Adding $\frac{x}{3}$ to Both Sides

When we add $\frac{x}{3}$ to both sides of the inequality, we get:

$${ 3 - \frac{x}{3} + \frac{x}{3} \ \textgreater \ -4 + \frac{x}{3} }$$

Simplifying the left-hand side, we get:

$${ 3 \ \textgreater \ -4 + \frac{x}{3} }$$

Subtracting 3 from Both Sides

To isolate the term involving $x$, we need to subtract 3 from both sides of the inequality. This will give us:

$${ 3 - 3 \ \textgreater \ -4 + \frac{x}{3} - 3 }$$

Simplifying the left-hand side, we get:

$${ 0 \ \textgreater \ -7 + \frac{x}{3} }$$

Adding 7 to Both Sides

To further isolate the term involving $x$, we need to add 7 to both sides of the inequality. This will give us:

$${ 0 + 7 \ \textgreater \ -7 + \frac{x}{3} + 7 }$$

Simplifying the left-hand side, we get:

$${ 7 \ \textgreater \ \frac{x}{3} }$$

Multiplying Both Sides by 3

To solve for $x$, we need to multiply both sides of the inequality by 3. This will give us:

$${ 7 \times 3 \ \textgreater \ \frac{x}{3} \times 3 }$$

Simplifying the left-hand side, we get:

$${ 21 \ \textgreater \ x }$$

Conclusion

In conclusion, the solution to the inequality ${ 3 - \frac{x}{3} \ \textgreater \ -4 }$ is $x \ \textless \ 21$. This means that any value of $x$ that is less than 21 will satisfy the given inequality.

Tips and Tricks

• When solving inequalities, it's essential to maintain the direction of the inequality sign.
• Adding or subtracting the same value to both sides of an inequality will not change the direction of the inequality.
• Multiplying or dividing both sides of an inequality by a negative value will reverse the direction of the inequality.

Real-World Applications

Inequalities have numerous real-world applications, including:

• Finance: Inequalities are used to calculate interest rates, investment returns, and loan payments.
• Science: Inequalities are used to model population growth, chemical reactions, and physical systems.
• Engineering: Inequalities are used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Final Thoughts

Solving inequalities requires a deep understanding of algebraic manipulation and inequality properties. By following the steps outlined in this article, you can solve complex inequalities and apply them to real-world problems. Remember to maintain the direction of the inequality sign and to check your solutions carefully.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Inequalities
• Mathway: Inequality Solver
• Wolfram Alpha: Inequality Solver

Frequently Asked Questions

Q: What is the solution to the inequality ${ 3 - \frac{x}{3} \ \textgreater \ -4 }$? A: The solution to the inequality is $x \ \textless \ 21$.

Q: How do I solve inequalities? A: To solve inequalities, you need to isolate the variable on one side of the inequality sign, while maintaining the direction of the inequality.

Q: What are some real-world applications of inequalities? A: Inequalities have numerous real-world applications, including finance, science, and engineering.

Introduction

In our previous article, we discussed how to solve the inequality ${ 3 - \frac{x}{3} \ \textgreater \ -4 }$. In this article, we will address some of the most frequently asked questions about inequalities, including their properties, solutions, and real-world applications.

Q&A

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

A: An equation is a mathematical statement that says two expressions are equal, while an inequality is a mathematical statement that says one expression is greater than, less than, or equal to another expression.

Q: How do I know which direction to write the inequality sign?

A: When solving an inequality, you need to determine the direction of the inequality sign based on the operation being performed. For example, if you are adding a positive value to both sides of the inequality, the direction of the inequality sign will remain the same. However, if you are multiplying or dividing both sides of the inequality by a negative value, the direction of the inequality sign will be reversed.

Q: Can I add or subtract the same value to both sides of an inequality?

A: Yes, you can add or subtract the same value to both sides of an inequality without changing the direction of the inequality sign.

Q: Can I multiply or divide both sides of an inequality by a negative value?

A: No, you cannot multiply or divide both sides of an inequality by a negative value without reversing the direction of the inequality sign.

Q: How do I solve an inequality with a fraction?

A: To solve an inequality with a fraction, you need to eliminate the fraction by multiplying both sides of the inequality by the denominator of the fraction.

Q: Can I use the same steps to solve a linear inequality as I would to solve a quadratic inequality?

A: No, you cannot use the same steps to solve a linear inequality as you would to solve a quadratic inequality. Linear inequalities can be solved using basic algebraic manipulations, while quadratic inequalities require more advanced techniques, such as factoring or the quadratic formula.

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

A: Inequalities have numerous real-world applications, including finance, science, and engineering. For example, inequalities can be used to calculate interest rates, investment returns, and loan payments in finance. In science, inequalities can be used to model population growth, chemical reactions, and physical systems. In engineering, inequalities can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Q: How do I check my solutions to an inequality?

A: To check your solutions to an inequality, you need to plug the solution back into the original inequality and verify that it is true.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality, but you need to be careful to enter the correct values and operations.

Tips and Tricks

• When solving inequalities, it's essential to maintain the direction of the inequality sign.
• Adding or subtracting the same value to both sides of an inequality will not change the direction of the inequality sign.
• Multiplying or dividing both sides of an inequality by a negative value will reverse the direction of the inequality sign.
• Inequalities can be used to model real-world problems, such as finance, science, and engineering.

Real-World Applications

Inequalities have numerous real-world applications, including:

• Finance: Inequalities can be used to calculate interest rates, investment returns, and loan payments.
• Science: Inequalities can be used to model population growth, chemical reactions, and physical systems.
• Engineering: Inequalities can be used to design and optimize systems, such as bridges, buildings, and electronic circuits.

Final Thoughts

Inequalities are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding how to solve inequalities, you can apply them to a wide range of problems, from finance and science to engineering and beyond.

Additional Resources

For further practice and review, we recommend the following resources:

• Khan Academy: Inequalities
• Mathway: Inequality Solver
• Wolfram Alpha: Inequality Solver

Frequently Asked Questions

Q: What is the difference between an inequality and an equation? A: An equation is a mathematical statement that says two expressions are equal, while an inequality is a mathematical statement that says one expression is greater than, less than, or equal to another expression.

Q: How do I know which direction to write the inequality sign? A: When solving an inequality, you need to determine the direction of the inequality sign based on the operation being performed.

Q: Can I add or subtract the same value to both sides of an inequality? A: Yes, you can add or subtract the same value to both sides of an inequality without changing the direction of the inequality sign.

Q: Can I multiply or divide both sides of an inequality by a negative value? A: No, you cannot multiply or divide both sides of an inequality by a negative value without reversing the direction of the inequality sign.

Q: How do I solve an inequality with a fraction? A: To solve an inequality with a fraction, you need to eliminate the fraction by multiplying both sides of the inequality by the denominator of the fraction.

Q: Can I use the same steps to solve a linear inequality as I would to solve a quadratic inequality? A: No, you cannot use the same steps to solve a linear inequality as you would to solve a quadratic inequality.

Q: What are some real-world applications of inequalities? A: Inequalities have numerous real-world applications, including finance, science, and engineering.

Q: How do I check my solutions to an inequality? A: To check your solutions to an inequality, you need to plug the solution back into the original inequality and verify that it is true.

Q: Can I use a calculator to solve an inequality? A: Yes, you can use a calculator to solve an inequality, but you need to be careful to enter the correct values and operations.