Solving $-4(2x-1) > 5-3x$ What Is The Correct First Step?

When faced with an inequality like $-4(2x-1) > 5-3x$, it's crucial to approach it systematically to arrive at the correct solution. Understanding the correct first step is paramount, as it lays the foundation for the subsequent steps. This article will dissect the inequality, explore the possible first steps, and explain why distributing -4 is the most appropriate initial action. We'll also delve into common pitfalls and alternative approaches, ensuring a comprehensive understanding of the process. The goal is to empower you with the knowledge and skills to confidently tackle similar inequalities. Let's embark on this mathematical journey together!

Understanding the Inequality

Before diving into the solution, it's essential to understand the anatomy of the inequality $-4(2x-1) > 5-3x$. Inequalities, unlike equations, involve comparing two expressions using symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). In this case, we have the 'greater than' symbol (>), indicating that the expression on the left side, $-4(2x-1)$, is greater than the expression on the right side, $5-3x$. Our mission is to isolate the variable 'x' to determine the range of values that satisfy this condition.

The left side of the inequality contains a parenthetical expression, $(2x-1)$, multiplied by -4. This immediately suggests the distributive property as a potential first step. The distributive property states that $a(b+c) = ab + ac$. Applying this property will eliminate the parentheses and simplify the expression. The right side of the inequality, $5-3x$, is a linear expression already in its simplest form. To successfully solve this inequality, we need to simplify both sides, combine like terms, and ultimately isolate 'x'. Failing to recognize the structure of the inequality and the potential application of the distributive property can lead to incorrect solutions. Therefore, a thorough understanding of the inequality's components is crucial for selecting the correct first step.

Why Distributing -4 is the Key First Step

The correct first step in solving the inequality $-4(2x-1) > 5-3x$ is to distribute the -4 across the terms within the parentheses. This involves multiplying -4 by both 2x and -1. The resulting expression becomes $-8x + 4$. Distributing is essential because it eliminates the parentheses, which is a crucial step in simplifying the inequality. By removing the parentheses, we free up the terms inside to interact with the other terms in the inequality, paving the way for combining like terms and isolating the variable 'x'.

Mathematically, distributing -4 is justified by the distributive property of multiplication over addition and subtraction. This property allows us to multiply a single term by a group of terms inside parentheses, effectively breaking down the expression into simpler components. In our case, $-4(2x-1)$ becomes $-4 * 2x + (-4) * (-1)$, which simplifies to $-8x + 4$. This transformation is not merely cosmetic; it's a fundamental algebraic manipulation that allows us to proceed with solving the inequality.

Consider the alternative approaches. Subtracting 2x from both sides (option C) or adding 1 to both sides (option D) are valid algebraic manipulations, but they don't address the immediate obstacle posed by the parentheses. These operations would leave the expression within the parentheses untouched, making further simplification more difficult. Therefore, while these steps might be useful later in the solution process, they are not the most efficient or logical first step. Only by distributing the -4 can we effectively simplify the left side of the inequality and prepare it for subsequent operations.

Option B, which incorrectly distributes -4 as $-8x - 1$, highlights the importance of careful application of the distributive property. The negative sign in front of the 1 within the parentheses must be considered when multiplying by -4. A mistake here can lead to an incorrect solution. Therefore, while the act of distributing is the correct first step, it must be executed with precision.

Step-by-Step Breakdown of Distributing -4

To illustrate the process of distributing -4 in the inequality $-4(2x-1) > 5-3x$, let's break it down into a step-by-step explanation:

1. Identify the term to be distributed: In this case, it's -4, which is multiplied by the expression within the parentheses, $(2x-1)$.
2. Apply the distributive property: Multiply -4 by each term inside the parentheses separately.
   • $-4 * 2x = -8x$
   • $-4 * -1 = +4$ (Remember, a negative times a negative is a positive)
3. Rewrite the inequality: Replace $-4(2x-1)$ with the result of the distribution, which is $-8x + 4$. The inequality now looks like this: $-8x + 4 > 5 - 3x$.

This seemingly simple process is the cornerstone of solving the inequality. By distributing the -4, we have transformed the left side of the inequality from a complex expression involving parentheses into a simpler linear expression. This sets the stage for the next steps in the solution, which will involve combining like terms and isolating 'x'. Without this initial distribution, the subsequent steps would be significantly more challenging, if not impossible. This careful and methodical approach ensures accuracy and efficiency in solving the inequality.

Common Mistakes and Pitfalls

When solving inequalities, several common mistakes can derail the process and lead to incorrect solutions. Recognizing these pitfalls is crucial for achieving accuracy. In the context of the inequality $-4(2x-1) > 5-3x$, one frequent error is incorrectly distributing the -4. As discussed earlier, the negative sign is often overlooked, leading to errors like $-4 * -1 = -4$ instead of the correct $+4$. This seemingly small mistake can drastically alter the solution.

Another common pitfall is forgetting to distribute to all terms within the parentheses. For example, a student might correctly multiply -4 by 2x to get -8x but then fail to multiply -4 by -1. This incomplete distribution leaves the expression within the parentheses partially intact, hindering further simplification.

Beyond the distribution step, other errors can occur during the solution process. One significant mistake is incorrectly applying the rules for multiplying or dividing inequalities by negative numbers. When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For instance, if we have $-2x > 4$, dividing both sides by -2 requires flipping the '>' sign to '<', resulting in $x < -2$. Forgetting this crucial rule is a common source of error.

Finally, careless arithmetic errors can plague even the most mathematically sound approaches. Simple addition or subtraction mistakes can propagate through the solution, leading to an incorrect answer. Therefore, it's essential to double-check each step and perform calculations meticulously.

To avoid these pitfalls, practice is paramount. Working through a variety of inequality problems, paying close attention to detail, and verifying each step will build confidence and accuracy. Remember, mathematics is a skill honed through consistent effort and mindful practice.

Alternative Approaches and Why They Aren't Ideal First Steps

While distributing -4 is the most efficient first step in solving the inequality $-4(2x-1) > 5-3x$, it's worth exploring alternative approaches to understand why they are less ideal as initial actions. One alternative, as suggested in option C, is to subtract 2x from both sides of the inequality. While this is a valid algebraic manipulation, it doesn't directly address the parentheses, which are the primary obstacle to simplifying the inequality. Subtracting 2x would result in $-4(2x-1) - 2x > 5 - 5x$. The parentheses remain, and the left side of the inequality becomes even more complex, making subsequent steps more challenging.

Another alternative, as suggested in option D, is to add 1 to both sides of the inequality. This action would yield $-4(2x-1) + 1 > 6 - 3x$. Again, the parentheses persist, and the overall complexity of the left side is not significantly reduced. Adding 1 doesn't bring us closer to isolating 'x' or simplifying the expression within the parentheses.

These alternative approaches are not inherently wrong; they are simply less efficient as initial steps. They don't tackle the core issue of the parentheses, which must be addressed to simplify the inequality effectively. Delaying the distribution of -4 prolongs the solution process and introduces unnecessary complexity.

The reason distributing -4 is the best first step is that it directly simplifies the expression by removing the parentheses. This simplification is crucial for combining like terms and ultimately isolating 'x'. Without this initial simplification, the other algebraic manipulations become more cumbersome and prone to errors. Therefore, while other steps might be necessary later in the solution, they are best performed after the distributive property has been applied.

Completing the Solution

Having established that distributing -4 is the correct first step, let's complete the solution of the inequality $-4(2x-1) > 5-3x$. We've already distributed the -4, resulting in the inequality $-8x + 4 > 5 - 3x$. Now, we need to isolate 'x'.

1. Combine like terms: Our goal is to get all the 'x' terms on one side of the inequality and the constant terms on the other side. Let's add 8x to both sides: $-8x + 4 + 8x > 5 - 3x + 8x$, which simplifies to $4 > 5 + 5x$.
2. Isolate the 'x' term: Subtract 5 from both sides: $4 - 5 > 5 + 5x - 5$, which simplifies to $-1 > 5x$.
3. Solve for 'x': Divide both sides by 5: $-1/5 > x$. This can also be written as $x < -1/5$.

Therefore, the solution to the inequality $-4(2x-1) > 5-3x$ is $x < -1/5$. This means that any value of 'x' less than -1/5 will satisfy the original inequality. It's important to note that because we divided by a positive number (5), we did not need to reverse the inequality sign.

This step-by-step completion of the solution demonstrates the power of starting with the correct first step. By distributing -4 initially, we set the stage for a smooth and logical progression towards the solution. Each subsequent step flowed naturally from the previous one, ultimately leading us to the correct answer. This illustrates the importance of strategic problem-solving in mathematics: a well-chosen first step can significantly simplify the entire process.

Conclusion

In conclusion, when tackling the inequality $-4(2x-1) > 5-3x$, the correct first step is undoubtedly to distribute the -4. This action eliminates the parentheses, simplifying the expression and paving the way for subsequent steps. We've explored the mathematical justification for this choice, highlighting the distributive property and its role in simplifying algebraic expressions. We've also examined alternative approaches and explained why they are less efficient as initial actions.

Understanding the correct first step is not just about memorizing a rule; it's about developing a strategic approach to problem-solving. By recognizing the structure of the inequality and identifying the key obstacle (the parentheses), we can make an informed decision about the most effective initial action. This approach not only leads to the correct solution but also fosters a deeper understanding of mathematical principles.

Furthermore, we've discussed common mistakes and pitfalls, emphasizing the importance of careful attention to detail and a thorough understanding of algebraic rules. Avoiding these errors is crucial for achieving accuracy in solving inequalities.

By mastering the art of solving inequalities, you equip yourself with a valuable tool for tackling a wide range of mathematical problems. So, embrace the challenge, practice diligently, and remember the importance of a well-chosen first step. Mathematics is a journey of discovery, and each solved problem is a step forward on that journey.