Solving $-2(8x - 4) < 2x + 5$ Inequality Step-by-Step

In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries. Solving inequalities is a fundamental skill, enabling us to determine the set of values that satisfy a given condition. This article delves into the step-by-step solution of the inequality $-2(8x - 4) < 2x + 5$, providing a clear and concise explanation for students and enthusiasts alike. We will explore the algebraic manipulations required to isolate the variable x and arrive at the correct solution. Understanding how to solve inequalities is essential not only for academic success but also for various real-world applications where constraints and limitations need to be considered. So, let’s embark on this mathematical journey and unravel the solution to this intriguing inequality.

Understanding the Inequality

Before we dive into the solution, let's first understand the anatomy of the inequality $-2(8x - 4) < 2x + 5$. This inequality states that the expression $-2(8x - 4)$ is strictly less than the expression $2x + 5$. Our goal is to find all values of x that make this statement true. Inequalities, unlike equations, often have a range of solutions rather than a single value. This range can be represented graphically on a number line or expressed in interval notation. The presence of the '<' symbol indicates that we are looking for values where the left-hand side is smaller than the right-hand side. To solve this inequality, we will employ algebraic techniques similar to those used for solving equations, but with a crucial difference: when multiplying or dividing both sides by a negative number, we must flip the direction of the inequality sign. This is a key concept to remember when working with inequalities.

Step 1: Distribute the -2

The first step in solving the inequality is to simplify both sides by performing any necessary operations. In this inequality, we need to distribute the -2 on the left side of the inequality. This means multiplying -2 by both terms inside the parentheses: $8x$ and -4. Let's break down the distribution:

-2 * (8x) = -16x -2 * (-4) = 8

So, after distributing the -2, the left side of the inequality becomes -16x + 8. Our inequality now looks like this:

-16x + 8 < 2x + 5

This step is crucial as it eliminates the parentheses and allows us to combine like terms in the subsequent steps. By correctly distributing the -2, we've simplified the inequality and brought it closer to a form where we can isolate the variable x. This process highlights the importance of careful arithmetic and attention to signs when manipulating algebraic expressions. The next step involves gathering the x terms on one side and the constant terms on the other, paving the way for isolating x and finding the solution.

Step 2: Move x Terms to One Side

Now that we've distributed the -2, our inequality is -16x + 8 < 2x + 5. The next step is to consolidate all the terms containing x on one side of the inequality. To do this, we can add 16x to both sides. Adding the same quantity to both sides of an inequality maintains the inequality, just as it does with equations. This operation will eliminate the -16x term on the left side:

-16x + 8 + 16x < 2x + 5 + 16x

Simplifying both sides, we get:

8 < 18x + 5

By adding 16x to both sides, we've successfully moved all the x terms to the right side of the inequality. This is a strategic move that brings us closer to isolating x. The inequality now has a simpler form, with the x term only appearing on the right side. This step demonstrates a fundamental principle in solving inequalities: performing the same operation on both sides preserves the inequality. With the x terms consolidated, the next step involves isolating the constant terms, which will further simplify the inequality and bring us closer to the solution.

Step 3: Move Constant Terms to the Other Side

Having moved all the x terms to the right side, our inequality now reads 8 < 18x + 5. The next logical step is to isolate the x term by moving all the constant terms to the left side. To achieve this, we subtract 5 from both sides of the inequality. Subtracting the same quantity from both sides maintains the inequality, just like addition. This operation will eliminate the +5 term on the right side:

8 - 5 < 18x + 5 - 5

Simplifying both sides, we get:

3 < 18x

By subtracting 5 from both sides, we've successfully isolated the term containing x on the right side and consolidated the constant terms on the left. This step is crucial because it prepares us for the final step of isolating x itself. The inequality is now in a form where we can easily see the relationship between the constant term and the x term. This process highlights the importance of performing inverse operations to isolate the variable of interest. With the constant terms now on one side, the final step is to divide both sides by the coefficient of x, which will reveal the solution to the inequality.

Step 4: Divide to Isolate x

Our inequality is currently 3 < 18x. To finally isolate x, we need to divide both sides of the inequality by the coefficient of x, which is 18. Dividing both sides by a positive number maintains the inequality, so we don't need to flip the inequality sign. Let's perform the division:

3 / 18 < 18x / 18

Simplifying both sides, we get:

1/6 < x

This inequality states that 1/6 is less than x, which is the same as saying x is greater than 1/6. We have successfully isolated x and found the solution to the inequality. This step demonstrates the final act of isolating the variable by using division, a fundamental operation in solving inequalities and equations alike. The solution x > 1/6 represents a range of values, meaning any value of x greater than 1/6 will satisfy the original inequality. This range can be visualized on a number line, where all points to the right of 1/6 (excluding 1/6 itself) are part of the solution set. With this final step, we've completely solved the inequality and determined the set of values that make it true.

The Solution

After performing the algebraic manipulations, we've arrived at the solution: x > 1/6. This inequality tells us that any value of x greater than 1/6 will satisfy the original inequality $-2(8x - 4) < 2x + 5$. The solution can be visualized on a number line as an open interval extending from 1/6 to positive infinity. The open interval indicates that 1/6 itself is not included in the solution set. In interval notation, the solution is represented as (1/6, ∞). Understanding the solution is crucial as it provides a clear range of values that make the inequality true. This solution has practical implications in various mathematical and real-world contexts, where constraints and limitations are expressed as inequalities. By solving the inequality, we've not only found the mathematical answer but also gained insight into the range of possible values that satisfy the given condition. The solution x > 1/6 is the final answer, encapsulating the result of our step-by-step algebraic journey.

Answer: A. x > rac{1}{6}

Therefore, the correct answer to the inequality $-2(8x - 4) < 2x + 5$ is A. x > rac{1}{6}. This confirms our step-by-step solution and highlights the importance of careful algebraic manipulation in arriving at the correct answer. The process of solving this inequality has reinforced the key concepts of distribution, combining like terms, and isolating the variable. Understanding these concepts is essential for tackling more complex inequalities and equations in mathematics. The solution x > rac{1}{6} represents a set of values that satisfy the given condition, demonstrating the power of inequalities in defining ranges and boundaries. This final answer solidifies our understanding of the problem and provides a clear and concise solution.