Solving 2(5-x) ≤ 14 A Step-by-Step Guide

In this article, we will walk through the process of solving the inequality $2(5-x) ${}$\leq 14$. Inequalities are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in algebra and beyond. We will break down each step, providing a clear explanation to ensure you grasp the underlying principles. By the end of this guide, you will not only know the correct answer but also understand the methodology behind it. So, let's dive in and solve this inequality together!

Understanding Inequalities

Before we tackle the specific problem at hand, let's briefly discuss what inequalities are and how they differ from equations. Inequalities, unlike equations, do not state that two expressions are equal. Instead, they indicate a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols used in inequalities are: > (greater than), < (less than), $\geq$ (greater than or equal to), and $\leq$ (less than or equal to). Understanding these symbols is the first step in solving inequalities. For instance, the expression $a > b$ means that a is greater than b, while $a \leq b$ means that a is less than or equal to b. This subtle difference in notation has significant implications for the solution process, particularly when dealing with negative numbers. When we multiply or divide both sides of an inequality by a negative number, we must reverse the direction of the inequality sign. This is a critical rule to remember, as neglecting it can lead to incorrect solutions. In our case, we are dealing with the inequality $2(5-x) \leq 14$, which involves the "less than or equal to" symbol. This means that our solution will include all values of x that make the expression $2(5-x)$ either less than or equal to 14. As we proceed, we will apply algebraic manipulations to isolate x and determine the range of values that satisfy this condition. The process will involve distributing, combining like terms, and potentially multiplying or dividing by negative numbers, so it's essential to keep the rules of inequalities in mind. Understanding these basic principles will not only help you solve this specific inequality but also equip you with the tools to tackle more complex problems in the future.

Step-by-Step Solution of 2(5-x) ≤ 14

Now, let's solve the inequality $2(5-x) \leq 14$ step by step. Our primary goal is to isolate the variable x on one side of the inequality. This will involve a series of algebraic manipulations, ensuring we maintain the integrity of the inequality throughout the process. The first step in solving this inequality is to distribute the 2 across the terms inside the parentheses. This means we multiply both 5 and -x by 2. Performing this operation, we get: $2 * 5 - 2 * x \leq 14$, which simplifies to $10 - 2x \leq 14$. Distributing is a crucial step as it removes the parentheses and allows us to combine like terms later on. Next, we want to isolate the term containing x. To do this, we need to subtract 10 from both sides of the inequality. This operation maintains the balance of the inequality and moves us closer to isolating x. Subtracting 10 from both sides, we have: $10 - 2x - 10 \leq 14 - 10$, which simplifies to $-2x \leq 4$. Now, we have the term -2x on the left side. To isolate x, we need to divide both sides by -2. Here's where we must remember a critical rule: when dividing or multiplying an inequality by a negative number, we must reverse the direction of the inequality sign. So, dividing both sides by -2, we get: $(-2x) / -2 \geq 4 / -2$. Notice that the “less than or equal to” sign ($\leq$) has changed to “greater than or equal to” sign ($\geq$). Simplifying the expression, we have: $x \geq -2$. This is our solution. It means that any value of x that is greater than or equal to -2 will satisfy the original inequality. To verify this, we can substitute a value greater than -2, such as 0, into the original inequality: $2(5 - 0) \leq 14$, which simplifies to $10 \leq 14$, a true statement. This confirms that our solution is correct. Understanding each step in this process is essential for solving more complex inequalities. Remember to distribute, combine like terms, and most importantly, reverse the inequality sign when multiplying or dividing by a negative number.

Identifying the Correct Answer

After solving the inequality $2(5-x) \leq 14$, we arrived at the solution $x \geq -2$. Now, let's compare this solution with the options provided to identify the correct answer. The options given are: A) $x \leq 2$ B) $x \geq 2$ C) $x \geq -2$ D) $x \leq 1$

By directly comparing our solution, $x \geq -2$, with the options, it's clear that option C) $x \geq -2$ matches our result. This means that the correct answer is C. The other options can be easily ruled out. Option A, $x \leq 2$, suggests that x is less than or equal to 2, which is the opposite of our solution. Option B, $x \geq 2$, indicates that x is greater than or equal to 2, which is a completely different range of values than our solution. Option D, $x \leq 1$, suggests that x is less than or equal to 1, which also does not align with our solution. To further solidify our understanding, we can visualize the solution on a number line. The inequality $x \geq -2$ represents all values of x that are -2 or greater. On a number line, this would be represented by a closed circle at -2 (indicating that -2 is included in the solution) and an arrow extending to the right, indicating all values greater than -2. This visual representation reinforces that option C is the correct answer. Identifying the correct answer is not just about finding a matching statement; it's about understanding the solution in the context of the original problem and verifying that it logically fits. In this case, our step-by-step solution and the comparison with the provided options clearly demonstrate that $x \geq -2$ is the accurate solution to the inequality $2(5-x) \leq 14$. This process of solving and verifying is crucial for building confidence and accuracy in mathematics.

Common Mistakes and How to Avoid Them

When solving inequalities, several common mistakes can lead to incorrect solutions. Recognizing these pitfalls and learning how to avoid them is crucial for mastering the topic. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. As we discussed earlier, this is a critical rule in inequality manipulation. For example, in our problem, we divided both sides of $-2x \leq 4$ by -2. If we had forgotten to reverse the inequality sign, we would have incorrectly arrived at $x \leq -2$, which is the opposite of the correct solution. To avoid this mistake, always double-check the sign of the number you are multiplying or dividing by. If it's negative, remember to flip the inequality sign. Another common mistake is incorrectly distributing a number across parentheses. For instance, in the initial step of our problem, we had to distribute 2 across (5 - x). An error here could involve only multiplying 2 by 5 and not by -x, leading to an incorrect equation. To prevent this, make sure you distribute the number to every term inside the parentheses. A helpful technique is to write out the distribution explicitly: 2 * 5 - 2 * x, which makes it less likely to overlook a term. Misunderstanding the inequality symbols themselves can also lead to errors. For example, confusing “greater than” (>) with “greater than or equal to” ($\geq$) can result in an incomplete or incorrect solution set. Remember that $\geq$ and $\leq$ include the endpoint in the solution, while > and < do not. To avoid this, take a moment to clarify the meaning of each symbol before proceeding with the problem. Arithmetic errors during the simplification process are also a common source of mistakes. Simple addition, subtraction, multiplication, or division errors can throw off the entire solution. To minimize these errors, work methodically, double-check your calculations, and consider using a calculator for complex arithmetic operations. Finally, failing to check your solution can leave you unaware of potential errors. After solving an inequality, it's a good practice to substitute a value from your solution set back into the original inequality to verify that it holds true. This can help you catch mistakes and build confidence in your answer. By being mindful of these common pitfalls and actively working to avoid them, you can significantly improve your accuracy and proficiency in solving inequalities.

Conclusion

In conclusion, we have successfully solved the inequality $2(5-x) \leq 14$ by following a step-by-step approach. We began by understanding the basics of inequalities and their properties. We then proceeded to distribute, isolate the variable x, and remember the crucial step of reversing the inequality sign when dividing by a negative number. This led us to the solution $x \geq -2$, which corresponds to option C. We also discussed common mistakes to avoid, such as forgetting to reverse the inequality sign, incorrectly distributing, misunderstanding inequality symbols, making arithmetic errors, and failing to check the solution. By understanding these potential pitfalls and actively working to prevent them, you can improve your accuracy and confidence in solving inequalities. The ability to solve inequalities is a fundamental skill in mathematics, with applications in various fields, including algebra, calculus, and real-world problem-solving. Mastering this skill requires practice and attention to detail, but the rewards are well worth the effort. We encourage you to continue practicing with different types of inequalities to further solidify your understanding. Remember to always double-check your work and verify your solutions. With consistent effort and a clear understanding of the underlying principles, you can confidently tackle any inequality problem that comes your way. This step-by-step guide has provided you with the tools and knowledge to solve the given inequality and similar problems. Keep practicing, and you'll become more proficient in no time!