Solve The Inequality:${ 4 - 7k - 3k \ \textless \ -6 }$

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. Inequalities are mathematical statements that compare two or more values, indicating that one value is greater than, less than, or equal to another value. In this article, we will focus on solving linear inequalities, which are inequalities that can be written in the form of a linear equation. We will use the given inequality $4 - 7k - 3k < -6$ as an example to demonstrate the steps involved in solving linear inequalities.

Understanding the Basics of Inequalities

Before we dive into solving the given inequality, let's understand the basics of inequalities. Inequalities can be classified into three main types:

• Linear Inequalities: These are inequalities that can be written in the form of a linear equation, i.e., $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants.
• Quadratic Inequalities: These are inequalities that involve a quadratic expression, i.e., $ax^2 + bx + c < d$ or $ax^2 + bx + c > d$, where $a$, $b$, $c$, and $d$ are constants.
• Absolute Value Inequalities: These are inequalities that involve absolute value expressions, i.e., $|x| < a$ or $|x| > a$, where $a$ is a constant.

Solving the Given Inequality

Now that we have a basic understanding of inequalities, let's focus on solving the given inequality $4 - 7k - 3k < -6$. To solve this inequality, we need to isolate the variable $k$ on one side of the inequality.

Step 1: Combine Like Terms

The first step in solving the inequality is to combine like terms. In this case, we have two terms with the variable $k$, which are $-7k$ and $-3k$. We can combine these terms by adding their coefficients, which gives us $-10k$.

4 - 7k - 3k < -6
4 - 10k < -6

Step 2: Add 6 to Both Sides

Next, we need to add 6 to both sides of the inequality to isolate the term with the variable $k$. This gives us:

4 - 10k + 6 < -6 + 6
10 - 10k < 0

Step 3: Subtract 10 from Both Sides

Now, we need to subtract 10 from both sides of the inequality to further isolate the term with the variable $k$. This gives us:

10 - 10k - 10 < 0 - 10
-10k < -20

Step 4: Divide Both Sides by -10

Finally, we need to divide both sides of the inequality by -10 to solve for the variable $k$. However, when we divide both sides of an inequality by a negative number, we need to reverse the direction of the inequality. This gives us:

-10k / -10 > -20 / -10
k > 2

Conclusion

In this article, we have demonstrated the steps involved in solving linear inequalities. We used the given inequality $4 - 7k - 3k < -6$ as an example to illustrate the process of solving linear inequalities. By following the steps outlined in this article, you should be able to solve linear inequalities with ease.

Tips and Tricks

Here are some tips and tricks to help you solve linear inequalities:

• Combine like terms: When solving an inequality, combine like terms to simplify the expression.
• Add or subtract the same value to both sides: When solving an inequality, add or subtract the same value to both sides to isolate the term with the variable.
• Reverse the direction of the inequality: When dividing both sides of an inequality by a negative number, reverse the direction of the inequality.
• Check your solution: When solving an inequality, check your solution by plugging it back into the original inequality.

Common Mistakes to Avoid

Here are some common mistakes to avoid when solving linear inequalities:

• Not combining like terms: Failing to combine like terms can make the inequality more difficult to solve.
• Not adding or subtracting the same value to both sides: Failing to add or subtract the same value to both sides can make the inequality more difficult to solve.
• Not reversing the direction of the inequality: Failing to reverse the direction of the inequality when dividing both sides by a negative number can result in an incorrect solution.
• Not checking your solution: Failing to check your solution by plugging it back into the original inequality can result in an incorrect solution.

Real-World Applications

Linear inequalities have numerous real-world applications, including:

• Finance: Linear inequalities are used to model financial situations, such as determining the minimum amount of money needed to invest in a particular stock.
• Science: Linear inequalities are used to model scientific situations, such as determining the maximum amount of a particular substance that can be present in a solution.
• Engineering: Linear inequalities are used to model engineering situations, such as determining the minimum amount of material needed to build a particular structure.

Conclusion

Introduction

In our previous article, we discussed the basics of solving linear inequalities and provided a step-by-step guide on how to solve the inequality $4 - 7k - 3k < -6$. In this article, we will answer some frequently asked questions about solving linear inequalities.

Q&A

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

A: A linear equation is an equation that can be written in the form of $ax + b = c$, where $a$, $b$, and $c$ are constants. A linear inequality, on the other hand, is an inequality that can be written in the form of $ax + b < c$ or $ax + b > c$, where $a$, $b$, and $c$ are constants.

Q: How do I know which direction to reverse the inequality when dividing both sides by a negative number?

A: When dividing both sides of an inequality by a negative number, you need to reverse the direction of the inequality. For example, if you have the inequality $x > 2$ and you divide both sides by $-3$, the resulting inequality would be $x < -\frac{2}{3}$.

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

A: No, you cannot use the same steps to solve quadratic inequalities as you would to solve linear inequalities. Quadratic inequalities involve a quadratic expression, which requires a different set of steps to solve.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, plug it back into the original inequality. If the solution satisfies the inequality, then it is correct.

Q: Can I use a calculator to solve linear inequalities?

A: Yes, you can use a calculator to solve linear inequalities. However, it's always a good idea to check your solution by plugging it back into the original inequality to ensure that it is correct.

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

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

• Not combining like terms
• Not adding or subtracting the same value to both sides
• Not reversing the direction of the inequality when dividing both sides by a negative number
• Not checking your solution by plugging it back into the original inequality

Q: How do I apply linear inequalities to real-world problems?

A: Linear inequalities can be applied to a wide range of real-world problems, including finance, science, and engineering. For example, you can use linear inequalities to determine the minimum amount of money needed to invest in a particular stock or to determine the maximum amount of a particular substance that can be present in a solution.

Conclusion

In conclusion, solving linear inequalities is a crucial skill that has numerous real-world applications. By following the steps outlined in this article and avoiding common mistakes, you should be able to solve linear inequalities with ease. Remember to combine like terms, add or subtract the same value to both sides, reverse the direction of the inequality, and check your solution. With practice and patience, you will become proficient in solving linear inequalities and be able to apply this skill to real-world problems.

Additional Resources

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

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

Practice Problems

Try solving the following linear inequalities:

• $2x + 3 < 5$
• $x - 2 > 3$
• $4x - 2 < 10$

Answer Key

• $x < 1$
• $x > 5$
• $x > 2.5$