Solve For K K K . − 1 ( K + 2 ) − 17 \textgreater − 8 -1(k+2)-17\ \textgreater \ -8 − 1 ( K + 2 ) − 17 \textgreater − 8

Introduction

Linear inequalities are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific linear inequality, $-1(k+2)-17 > -8$, and provide a step-by-step guide on how to approach it.

Understanding the Inequality

Before we dive into solving the inequality, let's break it down and understand what it means. The inequality is in the form of $ax + b > c$, where $a$, $b$, and $c$ are constants. In this case, $a = -1$, $b = -17$, and $c = -8$. The inequality states that the expression $-1(k+2)-17$ is greater than $-8$.

Step 1: Simplify the Left-Hand Side

To simplify the left-hand side of the inequality, we need to distribute the negative sign to the terms inside the parentheses. This gives us:

$$-1(k+2)-17 = -k-2-17 = -k-19$$

So, the inequality becomes:

$$-k-19 > -8$$

Step 2: Add 19 to Both Sides

To isolate the variable $k$, we need to get rid of the constant term on the left-hand side. We can do this by adding 19 to both sides of the inequality. This gives us:

$$-k-19+19 > -8+19$$

Simplifying both sides, we get:

$$-k > 11$$

Step 3: Multiply Both Sides by -1

To solve for $k$, we need to get rid of the negative sign in front of the variable. We can do this by multiplying both sides of the inequality by $-1$. However, when we multiply an inequality by a negative number, we need to reverse the direction of the inequality sign. This gives us:

$$k < -11$$

Conclusion

In this article, we solved the linear inequality $-1(k+2)-17 > -8$ using a step-by-step approach. We simplified the left-hand side, added 19 to both sides, and multiplied both sides by $-1$ to solve for $k$. The final solution is $k < -11$.

Tips and Tricks

When solving linear inequalities, it's essential to follow the order of operations and simplify the left-hand side before adding or subtracting constants. Additionally, when multiplying an inequality by a negative number, remember to reverse the direction of the inequality sign.

Real-World Applications

Linear inequalities have numerous real-world applications in fields such as economics, finance, and engineering. For example, in economics, linear inequalities can be used to model the relationship between variables such as supply and demand. In finance, linear inequalities can be used to determine the minimum or maximum value of an investment. In engineering, linear inequalities can be used to design and optimize systems.

Common Mistakes to Avoid

When solving linear inequalities, it's essential to avoid common mistakes such as:

• Not simplifying the left-hand side before adding or subtracting constants
• Not reversing the direction of the inequality sign when multiplying by a negative number
• Not checking the solution for extraneous solutions

Practice Problems

To practice solving linear inequalities, try the following problems:

1. Solve the inequality $2x+5 > 11$
2. Solve the inequality $x-3 > 7$
3. Solve the inequality $-2x-4 > 12$

Conclusion

Introduction

In our previous article, we provided a step-by-step guide on how to solve linear inequalities. However, we understand that sometimes, it's easier to learn through questions and answers. In this article, we will address some common questions and provide answers to help you better understand how to solve linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that can be written in the form of $ax + b > c$, where $a$, $b$, and $c$ are constants. Linear inequalities can be used to model real-world problems and can be solved using algebraic methods.

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

A: To simplify the left-hand side of a linear inequality, you need to distribute the negative sign to the terms inside the parentheses. For example, if you have the inequality $-1(k+2)-17 > -8$, you would simplify it to $-k-19 > -8$.

Q: What happens when I multiply an inequality by a negative number?

A: When you multiply an inequality by a negative number, you need to reverse the direction of the inequality sign. For example, if you have the inequality $-k > 11$ and you multiply both sides by $-1$, you would get $k < -11$.

Q: How do I check if my solution is extraneous?

A: To check if your solution is extraneous, you need to plug the solution back into the original inequality and see if it's true. If the solution is not true, then it's an extraneous solution and you need to discard it.

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

A: No, you cannot use the same steps to solve quadratic inequalities. Quadratic inequalities are more complex and require different methods to solve. However, the steps outlined in this article can be used as a starting point to solve quadratic inequalities.

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

A: Linear inequalities can be used to model real-world problems in fields such as economics, finance, and engineering. For example, in economics, linear inequalities can be used to model the relationship between variables such as supply and demand. In finance, linear inequalities can be used to determine the minimum or maximum value of an investment. In engineering, linear inequalities can be used to design and optimize systems.

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

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

• Not simplifying the left-hand side before adding or subtracting constants
• Not reversing the direction of the inequality sign when multiplying by a negative number
• Not checking the solution for extraneous solutions

Q: How can I practice solving linear inequalities?

A: You can practice solving linear inequalities by trying the following problems:

1. Solve the inequality $2x+5 > 11$
2. Solve the inequality $x-3 > 7$
3. Solve the inequality $-2x-4 > 12$

Conclusion

Solving linear inequalities is a crucial skill for students to master. By following the step-by-step approach outlined in this article and practicing with real-world problems, you can become proficient in solving linear inequalities. Remember to simplify the left-hand side, add or subtract constants, and multiply by negative numbers with caution. With practice and patience, you can become a master of solving linear inequalities.

Additional Resources

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

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

Practice Problems

Try the following problems to practice solving linear inequalities:

1. Solve the inequality $3x-2 > 5$
2. Solve the inequality $x+4 > 9$
3. Solve the inequality $-x-3 > 2$

Answer Key

1. $x > 7/3$
2. $x > 5$
3. $x < -5$