Best First Step To Solve -4x + 2/5 > 5/10

Solving inequalities, much like solving equations, requires a systematic approach to isolate the variable. In the given inequality, $-4x + \frac{2}{5} > \frac{5}{10}$, our primary goal is to isolate 'x' on one side of the inequality. To achieve this, we must carefully consider the order of operations and the properties of inequalities. Understanding the best initial step is crucial for efficient and accurate problem-solving. Let's delve into the options provided and analyze why one stands out as the most logical first step.

Understanding the Inequality $-4x + \frac{2}{5} > \frac{5}{10}$

Before diving into the answer choices, let's break down the inequality itself. We have a variable term, $-4x$, a fractional constant, $\frac{2}{5}$, and another fraction, $\frac{5}{10}$. The inequality symbol, '>', indicates that the expression on the left side, $-4x + \frac{2}{5}$, is greater than the fraction on the right side, $\frac{5}{10}$. Remember that $\frac{5}{10}$ can be simplified to $\frac{1}{2}$, which might make calculations easier later on. The key here is to isolate the term with 'x' by systematically eliminating other terms on the same side. This process is guided by the order of operations in reverse, meaning we typically address addition and subtraction before multiplication and division.

When solving inequalities, it's crucial to remember one fundamental rule: multiplying or dividing both sides by a negative number requires flipping the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line. For example, if 2 < 4, then multiplying both sides by -1 gives -2 > -4. Failing to account for this rule will lead to an incorrect solution. In our case, the coefficient of 'x' is -4, a negative number, so we must be mindful of this rule when we eventually multiply or divide by -4.

To effectively isolate 'x', we need to undo the operations that are being performed on it. In the inequality $-4x + \frac{2}{5} > \frac{5}{10}$, 'x' is being multiplied by -4, and $\frac{2}{5}$ is being added to the result. Following the reverse order of operations, we should address the addition/subtraction before the multiplication/division. This means we should first deal with the $\frac{2}{5}$ term. The logical next step is to eliminate this term from the left side of the inequality, which will bring us closer to isolating 'x'. This strategic approach sets the stage for the subsequent steps and ensures a clear path toward the solution.

Analyzing the Answer Choices

Now, let's carefully examine the given answer choices to determine the most appropriate first step:

A. Add $\frac{2}{5}$ to both sides. B. Subtract $\frac{2}{5}$ from both sides. C. Multiply both sides by -4 and reverse the inequality symbol. D. Divide both sides by 10.

Option A: Add $\frac{2}{5}$ to Both Sides

Adding $\frac{2}{5}$ to both sides of the inequality would result in: $-4x + \frac{2}{5} + \frac{2}{5} > \frac{5}{10} + \frac{2}{5}$. While this is a valid algebraic manipulation, it doesn't help us isolate 'x'. Instead, it complicates the left side further by adding another term. Therefore, this option is not the most efficient first step.

Option B: Subtract $\frac{2}{5}$ from Both Sides

Subtracting $\frac{2}{5}$ from both sides of the inequality gives us: $-4x + \frac{2}{5} - \frac{2}{5} > \frac{5}{10} - \frac{2}{5}$. This action cancels out the $\frac{2}{5}$ term on the left side, moving us closer to isolating the term with 'x'. This is a standard technique in solving equations and inequalities, where we perform the inverse operation to eliminate terms. By subtracting $\frac{2}{5}$, we simplify the inequality and pave the way for the next steps in solving for 'x'. This option aligns with the strategy of addressing addition and subtraction before multiplication and division.

Option C: Multiply Both Sides by -4 and Reverse the Inequality Symbol

Multiplying both sides by -4 is a step we might need to take eventually, but it's not the most strategic first step. This is because it directly addresses the coefficient of 'x' before dealing with the constant term. While mathematically sound if we remember to reverse the inequality symbol, it introduces complexity prematurely. It's generally more efficient to isolate the variable term before dealing with its coefficient. Furthermore, performing this step first would result in fractions on the right side of the inequality, potentially making subsequent calculations more cumbersome. Therefore, while this operation is part of the overall solution, it's best left for later in the process.

Option D: Divide Both Sides by 10

Dividing both sides by 10 is not a relevant operation in this particular inequality. The 10 is only present in the denominator of the fraction $\frac{5}{10}$ on the right side. Dividing both sides by 10 wouldn't directly help in isolating 'x' and would instead create more complex fractions. This option doesn't align with the goal of simplifying the inequality and isolating the variable. It's a distraction from the core steps required to solve for 'x'.

The Best First Step: Subtract $\frac{2}{5}$ from Both Sides

Based on our analysis, the best first step in solving the inequality $-4x + \frac{2}{5} > \frac{5}{10}$ is B. Subtract $\frac{2}{5}$ from both sides. This step simplifies the inequality by eliminating the constant term on the left side, bringing us closer to isolating the term with 'x'. It follows the principle of addressing addition and subtraction before multiplication and division, which is a fundamental strategy in solving equations and inequalities. By subtracting $\frac{2}{5}$, we set the stage for a more straightforward solution process.

Detailed Steps After the First Step

To further illustrate the solution process, let's outline the subsequent steps after subtracting $\frac{2}{5}$ from both sides:

1. Subtract $\frac{2}{5}$ from both sides:

   $-4x + \frac{2}{5} - \frac{2}{5} > \frac{5}{10} - \frac{2}{5}$

   This simplifies to: $-4x > \frac{5}{10} - \frac{2}{5}$

2. Simplify the right side:

   To subtract the fractions, we need a common denominator. The least common denominator for 10 and 5 is 10. So, we convert $\frac{2}{5}$ to $\frac{4}{10}$:

   $-4x > \frac{5}{10} - \frac{4}{10}$

   $-4x > \frac{1}{10}$

3. Divide both sides by -4 and reverse the inequality symbol:

   Since we are dividing by a negative number, we must reverse the inequality sign:

   $\frac{-4x}{-4} < \frac{\frac{1}{10}}{-4}$

   $x < -\frac{1}{40}$

Therefore, the solution to the inequality is $x < -\frac{1}{40}$. This detailed step-by-step process highlights the importance of the initial step in setting the stage for an efficient and accurate solution. Choosing the correct first step, such as subtracting $\frac{2}{5}$ in this case, streamlines the process and minimizes the chances of errors.

Key Takeaways for Solving Inequalities

Solving inequalities requires a methodical approach, and choosing the right first step can significantly impact the ease and accuracy of the solution. Here are some key takeaways to keep in mind:

• Isolate the variable: The primary goal is to isolate the variable on one side of the inequality.
• Reverse order of operations: Follow the reverse order of operations (addition/subtraction before multiplication/division) when isolating the variable.
• Inverse operations: Use inverse operations to eliminate terms (e.g., subtract to undo addition).
• Negative multiplication/division: Remember to flip the inequality sign when multiplying or dividing both sides by a negative number.
• Simplify fractions: Simplify fractions whenever possible to make calculations easier.
• Strategic thinking: Choose the first step that most efficiently moves you towards isolating the variable.

By applying these principles, you can confidently tackle a wide range of inequality problems. In the case of $-4x + \frac{2}{5} > \frac{5}{10}$, subtracting $\frac{2}{5}$ from both sides is the optimal first step, paving the way for a clear and concise solution.

Conclusion

In conclusion, when faced with the inequality $-4x + \frac{2}{5} > \frac{5}{10}$, the best first step is to subtract $\frac{2}{5}$ from both sides (Option B). This action strategically isolates the term containing the variable 'x' and aligns with the fundamental principles of solving inequalities. By understanding the underlying concepts and applying a systematic approach, you can confidently navigate inequality problems and arrive at the correct solution. Remember to always consider the order of operations, inverse operations, and the critical rule of flipping the inequality sign when multiplying or dividing by a negative number. This thorough understanding will empower you to solve complex mathematical problems with accuracy and efficiency.