Solving Inequalities The Crucial First Step In -4(3-5x) ≥ -6x + 9
Navigating the world of inequalities can feel like traversing a complex maze, but with the right approach, even the most daunting problems can be solved. When confronted with an inequality like -4(3-5x) ≥ -6x + 9, the initial step is paramount. It sets the stage for the entire solution process, and a misstep here can lead to an incorrect answer. To master this crucial first step, we'll dissect the inequality, explore the order of operations, and identify the correct path forward.
Understanding the Inequality -4(3-5x) ≥ -6x + 9
The inequality -4(3-5x) ≥ -6x + 9 presents a relationship between two expressions. The symbol "≥" signifies "greater than or equal to," indicating that the expression on the left side, -4(3-5x), is either larger than or the same as the expression on the right side, -6x + 9. Our goal is to find the values of x that satisfy this condition. This requires us to isolate x on one side of the inequality, but before we can do that, we must simplify the expressions on both sides.
The left-hand side of the inequality, -4(3-5x), contains a product of a constant, -4, and a binomial expression, (3-5x). To simplify this, we need to apply the distributive property. This property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our case, we distribute -4 to both terms inside the parentheses: -4 * 3 and -4 * -5x. This is where the critical first step lies – ensuring we correctly apply the distributive property while paying close attention to the signs.
The Significance of the Distributive Property
The distributive property is a fundamental concept in algebra. It allows us to simplify expressions by multiplying a single term by multiple terms within parentheses. In the context of inequalities and equations, proper distribution is essential for maintaining the balance and arriving at a correct solution. When dealing with negative numbers, as in our example, the distributive property becomes even more crucial. A sign error during this step can propagate through the rest of the solution, leading to a wrong answer.
Let's examine the correct application of the distributive property to the expression -4(3-5x). We multiply -4 by 3, which gives us -12. Then, we multiply -4 by -5x. Here's where the sign is critical: a negative times a negative results in a positive. So, -4 * -5x = 20x. Therefore, the correct distribution yields -12 + 20x. Now, let's consider the incorrect options and understand why they are wrong.
Analyzing the Incorrect Options
The provided options present different variations of the distribution, highlighting common mistakes students make. Option A, -12-20x ≤ -6x + 9, incorrectly distributes the -4 by not recognizing that -4 multiplied by -5x results in a positive term. This is a classic error of overlooking the sign change when multiplying two negative numbers. Similarly, option B, -12-20x ≥ -6x + 9, makes the same mistake. The critical difference between these incorrect options and the correct one lies in the sign of the term involving x.
Option C, -12+20x ≤ -6x + 9, correctly handles the distribution of -4 to 3, resulting in -12. It also correctly multiplies -4 by -5x to get +20x. However, this option presents an inequality with a "less than or equal to" symbol (≤), whereas the original inequality uses "greater than or equal to" (≥). While the distribution is correct, the inequality sign is flipped, making this option incorrect as a first step in solving the original inequality. It's crucial to maintain the integrity of the inequality symbol throughout the initial steps of the solution.
The Correct First Step: D. -12 + 20x ≥ -6x + 9
Option D, -12 + 20x ≥ -6x + 9, stands out as the correct first step. It accurately applies the distributive property, transforming -4(3-5x) into -12 + 20x. Furthermore, it preserves the original inequality symbol, maintaining the relationship "greater than or equal to." This option demonstrates a clear understanding of both the distributive property and the importance of sign conventions in algebraic manipulations.
By correctly distributing the -4, we have simplified the left side of the inequality. The resulting expression, -12 + 20x, is now ready for further manipulation. We can proceed by isolating the x terms on one side and the constant terms on the other. This typically involves adding or subtracting terms from both sides of the inequality, always ensuring we maintain the balance and the direction of the inequality. For example, we might add 6x to both sides and then add 12 to both sides to further isolate x.
Continuing the Solution Process
Once we've established the correct first step, -12 + 20x ≥ -6x + 9, we can continue solving for x. The next logical step would be to add 6x to both sides of the inequality. This gives us -12 + 26x ≥ 9. By adding 6x, we move the x term from the right side to the left side, bringing us closer to isolating x. The key here is to perform the same operation on both sides to maintain the inequality's balance. Just as with equations, whatever you do to one side of an inequality, you must do to the other.
Following this, we would add 12 to both sides of the inequality to isolate the term with x. This leads to 26x ≥ 21. Adding 12 to both sides effectively cancels out the -12 on the left side and combines the constant terms on the right side. This step further simplifies the inequality, making it easier to solve for x. We are now one step away from finding the solution set for x.
Finally, to isolate x completely, we divide both sides of the inequality by 26. This yields x ≥ 21/26. Dividing both sides by 26 gives us the value of x that satisfies the inequality. Because we are dividing by a positive number, the direction of the inequality remains unchanged. If we were to divide by a negative number, we would need to flip the inequality sign. The solution x ≥ 21/26 represents all values of x that are greater than or equal to 21/26.
The Importance of Maintaining Inequality Direction
A critical concept in solving inequalities is understanding when and why to flip the inequality sign. As mentioned earlier, when you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality. This is because multiplying or dividing by a negative number changes the sign of the values, effectively flipping the number line. For example, if 2 < 4, multiplying both sides by -1 gives -2 > -4. Failing to flip the inequality sign when necessary is a common mistake that leads to incorrect solutions.
In our problem, we divided by a positive number (26), so the direction of the inequality remained unchanged. However, it's essential to be vigilant and remember this rule when solving other inequalities. Always consider the sign of the number you are multiplying or dividing by and make the appropriate adjustment to the inequality sign.
Conclusion: Mastering the First Step and Beyond
In conclusion, the correct first step in solving the inequality -4(3-5x) ≥ -6x + 9 is D. -12 + 20x ≥ -6x + 9. This step demonstrates the accurate application of the distributive property and a careful consideration of sign conventions. By mastering this initial step, we pave the way for a successful solution. However, it's crucial to remember that this is just the beginning. The subsequent steps, including isolating x and maintaining the correct inequality direction, are equally important.
Solving inequalities is a fundamental skill in mathematics, with applications in various fields, including algebra, calculus, and real-world problem-solving. By understanding the underlying principles and practicing diligently, you can confidently navigate the world of inequalities and arrive at accurate solutions. The journey begins with a single step, and in this case, that step is the correct application of the distributive property.
