Solving Inequalities: Find K In $-6(1+7k) + 7(1+6k) leq -2$
Are you grappling with algebraic inequalities? Let's break down how to solve the inequality $-6(1+7k) + 7(1+6k) \leq -2$ step by step. This comprehensive guide will walk you through the process, ensuring you understand each stage involved in isolating the variable k. Whether you're a student tackling homework or just brushing up on your algebra skills, this detailed explanation will help you master solving similar problems. This exploration into solving inequalities not only reinforces your understanding of algebraic manipulation but also highlights the importance of careful arithmetic and logical progression. So, let’s get started and unravel this inequality together!
Understanding the Inequality
Before we dive into the solution, let's ensure we understand what the inequality $-6(1+7k) + 7(1+6k) \leq -2$ represents. Inequalities are mathematical statements that compare two expressions using symbols like less than (<), greater than (>), less than or equal to (), and greater than or equal to (). In this case, we have a linear inequality because the highest power of the variable k is 1. Solving an inequality involves finding the range of values for k that make the statement true.
To begin, our key objective is to isolate k on one side of the inequality. This process involves several algebraic manipulations, including distributing, combining like terms, and performing operations on both sides of the inequality. It’s crucial to remember that multiplying or dividing both sides by a negative number will reverse the direction of the inequality sign. This is a fundamental rule in inequality manipulation, and overlooking it can lead to an incorrect solution. As we proceed, we'll highlight each step, explaining the rationale behind it and emphasizing any potential pitfalls to avoid. Understanding these foundational principles is vital not just for this specific problem but for tackling a wide array of algebraic challenges.
Step-by-Step Solution
Let's solve the inequality $-6(1+7k) + 7(1+6k) \leq -2$:
1. Distribute the constants:
Our first step is to distribute the constants outside the parentheses to the terms inside. This means multiplying -6 by both 1 and 7k, and multiplying 7 by both 1 and 6k. This is a crucial step in simplifying the inequality and making it easier to work with. Distributive property is a fundamental concept in algebra and is essential for solving equations and inequalities.
This simplifies to:
2. Combine Like Terms:
Next, we combine like terms on the left side of the inequality. Like terms are terms that have the same variable raised to the same power. In this case, we have two constant terms (-6 and +7) and two terms with the variable k (-42k and +42k). Combining like terms helps to further simplify the inequality and move us closer to isolating k.
Combining the constant terms (-6 + 7) gives us 1. Combining the k terms (-42k + 42k) gives us 0. Notice that the k terms cancel each other out, which is a significant observation. This cancellation will lead to a specific type of solution, which we will discuss further.
So, the inequality becomes:
3. Analyze the Result:
Now we have the simplified inequality $1 \leq -2$. This statement says that 1 is less than or equal to -2. This is a false statement, as 1 is clearly greater than -2. When solving inequalities, if we arrive at a statement that is always false, it means there is no solution to the inequality.
In other words, there is no value of k that we can substitute into the original inequality that will make the inequality true. This outcome is important to recognize, as it signifies that the inequality has no solution. It’s a different scenario from finding a specific value or range of values for the variable; instead, we've determined that the inequality is fundamentally not valid.
4. State the Solution:
Since the statement $1 \leq -2$ is false, the inequality has no solution. This means that no matter what value we substitute for k, the inequality will never be true. This result is a crucial part of the problem-solving process, indicating that the initial inequality is, in a sense, contradictory.
In mathematical notation, we can express this as an empty set, often denoted by the symbol ∅. This symbol signifies that there are no elements (in this case, values of k) that satisfy the condition set by the inequality. Recognizing when an inequality has no solution is as important as finding a solution, as it provides a complete understanding of the problem.
Common Mistakes to Avoid
When solving inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them:
1. Forgetting to Flip the Inequality Sign:
One of the most frequent errors is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. As mentioned earlier, this is a critical rule in inequality manipulation. If you multiply or divide by a negative value, you must reverse the direction of the inequality to maintain the truth of the statement. For instance, if you have -2k < 4, dividing both sides by -2 requires you to change the inequality to k > -2.
2. Incorrectly Distributing:
Incorrectly distributing constants across parentheses is another common mistake. Ensure that you multiply the constant by every term inside the parentheses. For example, in the expression -3(2x - 1), you need to multiply -3 by both 2x and -1, resulting in -6x + 3. A failure to distribute correctly can alter the entire equation and lead to a wrong solution.
3. Combining Unlike Terms:
Make sure to only combine like terms. Like terms have the same variable raised to the same power. For example, you can combine 3x and 5x to get 8x, but you cannot combine 3x and 5x². Mixing unlike terms will result in an incorrect simplification of the expression.
4. Arithmetic Errors:
Simple arithmetic errors can also derail your solution. Double-check your calculations, especially when dealing with negative numbers and fractions. A small mistake in addition, subtraction, multiplication, or division can propagate through the rest of the problem, leading to an incorrect answer. It’s often helpful to rework the steps and verify your calculations to catch any potential errors.
5. Misinterpreting the Solution:
Finally, misinterpreting the solution can be a problem, especially when the inequality has no solution or an infinite number of solutions. As we saw in our example, it's important to recognize when an inequality leads to a false statement, indicating no solution. Similarly, if you end up with a true statement (e.g., 0 < 2), it means that all real numbers are solutions. Understanding how to interpret these different outcomes is crucial for correctly answering the problem.
By being mindful of these common mistakes and taking the time to carefully review your work, you can increase your accuracy and confidence in solving inequalities.
Conclusion
In summary, we've walked through the process of solving the inequality $-6(1+7k) + 7(1+6k) \leq -2$. We began by distributing and simplifying the inequality, which led us to the statement $1 \leq -2$. Recognizing that this statement is false, we concluded that the inequality has no solution. This exercise highlights the importance of careful algebraic manipulation and the ability to interpret the results of those manipulations.
Solving inequalities is a fundamental skill in algebra and is essential for various applications in mathematics and real-world problem-solving. By understanding the steps involved and being aware of common mistakes, you can confidently tackle a wide range of inequality problems. Remember to pay close attention to the rules of inequality manipulation, especially when dealing with negative numbers, and always double-check your work to ensure accuracy.
By mastering these techniques, you'll be well-equipped to handle more complex algebraic challenges and develop a strong foundation in mathematical problem-solving. For further learning and practice, consider exploring resources like Khan Academy's algebra section, which offers numerous lessons and exercises on inequalities and other algebraic topics.
