Solving The Inequality -2(3x+6) ≥ 4(x+7) A Step-by-Step Guide

In the realm of mathematics, inequalities play a crucial role in defining relationships between quantities that are not necessarily equal. They provide a framework for expressing a range of possible values that satisfy a given condition. This article delves into the intricacies of solving the inequality $-2(3x+6) \geq 4(x+7)$, providing a comprehensive step-by-step guide to arrive at the correct solution. Our primary goal is to identify the inequality that accurately represents all possible solutions for the variable x. We will meticulously dissect each step, ensuring clarity and understanding for readers of all backgrounds.

Understanding the Inequality

Before we embark on the solution process, let's first grasp the essence of the given inequality: $-2(3x+6) \geq 4(x+7)$. This mathematical statement asserts that the expression on the left-hand side, $-2(3x+6)$, is greater than or equal to the expression on the right-hand side, $4(x+7)$. Our mission is to isolate the variable x and determine the range of values that satisfy this condition. This involves a series of algebraic manipulations, each designed to bring us closer to the solution. Understanding the properties of inequalities is paramount. Multiplying or dividing both sides by a negative number requires flipping the inequality sign, a critical detail that we will emphasize throughout our solution.

Step-by-Step Solution

1. Distribute the Constants

The initial step in solving this inequality involves distributing the constants on both sides of the equation. This means multiplying the constants outside the parentheses by each term inside the parentheses. Let's break it down:

• On the left-hand side, we have $-2(3x+6)$. Distributing the -2, we get: $-2 * 3x + (-2) * 6 = -6x - 12$.
• On the right-hand side, we have $4(x+7)$. Distributing the 4, we get: $4 * x + 4 * 7 = 4x + 28$.

Now, our inequality looks like this: $-6x - 12 \geq 4x + 28$. This simplification sets the stage for the next steps in isolating the variable x. This step is crucial as it removes the parentheses, allowing us to combine like terms and further simplify the inequality. Accurate distribution is key to avoiding errors in the subsequent steps.

2. Combine Like Terms

To isolate the variable x, we need to gather all terms containing x on one side of the inequality and all constant terms on the other side. We can achieve this by performing algebraic manipulations that maintain the balance of the inequality. Let's move the x terms to the left-hand side and the constants to the right-hand side.

• Add 6x to both sides: $-6x - 12 + 6x \geq 4x + 28 + 6x$. This simplifies to $-12 \geq 10x + 28$.
• Subtract 28 from both sides: $-12 - 28 \geq 10x + 28 - 28$. This simplifies to $-40 \geq 10x$.

Now, our inequality is in a much simpler form: $-40 \geq 10x$. This step has consolidated the x terms and the constant terms, making it easier to isolate x in the next step. Careful attention to the signs is crucial during this process to ensure accuracy.

3. Isolate the Variable

We're now in the final stretch of isolating x. To do this, we need to divide both sides of the inequality by the coefficient of x, which is 10 in this case. Remember the crucial rule: if we divide or multiply both sides of an inequality by a negative number, we must flip the inequality sign.

• Divide both sides by 10: $-40 / 10 \geq 10x / 10$. This simplifies to $-4 \geq x$.

This inequality, $-4 \geq x$, is equivalent to saying that x is less than or equal to -4. We can rewrite this as $x \leq -4$ to match the standard notation. This step directly reveals the solution set for the inequality. Dividing by a positive number maintains the direction of the inequality, which is why we didn't flip the sign in this case.

The Solution

Therefore, the solution to the inequality $-2(3x+6) \geq 4(x+7)$ is $x \leq -4$. This means that any value of x that is less than or equal to -4 will satisfy the original inequality. We have successfully navigated through the algebraic manipulations to arrive at this solution. The solution set encompasses all real numbers less than or equal to -4, representing a continuous range of values.

Analyzing the Answer Choices

Now, let's examine the answer choices provided and identify the one that matches our solution:

A. $x \geq -4$ B. $x \leq -4$ C. $x \geq 8$ D. $x \leq 8$

Comparing our solution, $x \leq -4$, with the answer choices, we can clearly see that option B, $x \leq -4$, is the correct answer. The other options represent different ranges of values and do not satisfy the original inequality.

Importance of Understanding Inequalities

Inequalities are fundamental tools in mathematics, used to model and solve a wide range of real-world problems. They appear in various fields, including economics, physics, and computer science. Mastering the techniques for solving inequalities is essential for success in higher-level mathematics and related disciplines. This example highlights the importance of understanding the properties of inequalities and applying algebraic manipulations correctly.

Conclusion

In conclusion, we have successfully solved the inequality $-2(3x+6) \geq 4(x+7)$ and determined that the correct answer is $x \leq -4$. This process involved distributing constants, combining like terms, and isolating the variable x. We emphasized the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. This step-by-step approach provides a clear understanding of the solution process, empowering readers to tackle similar problems with confidence. Understanding inequalities is a cornerstone of mathematical literacy, enabling us to analyze and solve a wide array of problems.

By meticulously following each step and understanding the underlying principles, we can confidently solve inequalities and apply them to real-world scenarios. This example serves as a testament to the power of algebraic manipulation and the importance of attention to detail in mathematical problem-solving. Practice and consistent application of these techniques will solidify your understanding and enhance your problem-solving skills.