Solve The Following Inequality For $k$. Write Your Answer In Simplest Form.$7 + 5(-4k - 10) \ \textgreater \ -9k + 2 - 8$k \ \textless \ \square$

Introduction

In mathematics, inequalities are a fundamental concept that plays a crucial role in solving various problems. An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other. In this article, we will focus on solving a specific inequality for the variable $k$. We will break down the solution into manageable steps, making it easier to understand and follow.

The Given Inequality

The given inequality is:

$$7 + 5(-4k - 10) \ \textgreater \ -9k + 2 - 8$$

Our goal is to solve this inequality for $k$ and express the solution in simplest form.

Step 1: Distribute and Simplify

To begin solving the inequality, we need to distribute the negative sign to the terms inside the parentheses and simplify the expression.

$$7 + 5(-4k - 10) \ \textgreater \ -9k + 2 - 8$$

$$7 - 20k - 50 \ \textgreater \ -9k + 2 - 8$$

Combine like terms:

$$-20k - 43 \ \textgreater \ -9k - 6$$

Step 2: Isolate the Variable

Next, we need to isolate the variable $k$ on one side of the inequality. To do this, we will add $9k$ to both sides of the inequality and then subtract $43$ from both sides.

$$-20k - 43 + 9k \ \textgreater \ -9k - 6 + 9k$$

$$-11k - 43 \ \textgreater \ -6$$

Add $43$ to both sides:

$$-11k - 43 + 43 \ \textgreater \ -6 + 43$$

$$-11k \ \textgreater \ 37$$

Step 3: Solve for $k$

Now that we have isolated the variable $k$, we can solve for it by dividing both sides of the inequality by $-11$.

$$\frac{-11k}{-11} \ \textgreater \ \frac{37}{-11}$$

$$k \ \textless \ -\frac{37}{11}$$

Conclusion

In this article, we solved the given inequality for the variable $k$. We broke down the solution into manageable steps, making it easier to understand and follow. By distributing and simplifying the expression, isolating the variable, and solving for $k$, we arrived at the solution:

$$k \ \textless \ -\frac{37}{11}$$

This solution indicates that the value of $k$ must be less than $-\frac{37}{11}$.

Final Answer

Q&A: Solving Inequalities for $k$

Q: What is an inequality?

A: An inequality is a statement that compares two expressions, indicating that one is greater than, less than, or equal to the other.

Q: How do I solve an inequality for a variable?

A: To solve an inequality for a variable, you need to isolate the variable on one side of the inequality. This can be done by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides by the same non-zero value.

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

A: Solving an inequality is similar to solving an equation, but with one key difference: the solution to an inequality is a range of values, rather than a single value. This means that the solution to an inequality is often expressed as an interval, rather than a single number.

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

A: The direction of the inequality is determined by the sign of the coefficient of the variable. If the coefficient is positive, the inequality is "greater than" or "less than". If the coefficient is negative, the inequality is "less than" or "greater than".

Q: What is the solution to the inequality $7 + 5(-4k - 10) \ \textgreater \ -9k + 2 - 8$?

A: The solution to the inequality $7 + 5(-4k - 10) \ \textgreater \ -9k + 2 - 8$ is $k \ \textless \ -\frac{37}{11}$.

Q: How do I express the solution to an inequality in simplest form?

A: To express the solution to an inequality in simplest form, you need to simplify the expression on both sides of the inequality. This may involve combining like terms, canceling out common factors, or rewriting the inequality in a different form.

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

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

• Not distributing the negative sign to the terms inside the parentheses
• Not combining like terms
• Not isolating the variable on one side of the inequality
• Not checking the direction of the inequality
• Not expressing the solution in simplest form

Conclusion

In this article, we provided a step-by-step guide to solving inequalities for the variable $k$. We also answered some common questions about solving inequalities, including how to determine the direction of the inequality, how to express the solution in simplest form, and what common mistakes to avoid. By following these steps and avoiding these common mistakes, you can become more confident and proficient in solving inequalities.

Final Answer

The final answer is $\boxed{-\frac{37}{11}}$.