Solving Inequalities A Step-by-Step Guide To 2(4+2x) ≥ 5x + 5

In this article, we will delve into the world of inequalities, focusing on the specific inequality $2(4 + 2x) \geq 5x + 5$. Inequalities, similar to equations, are mathematical statements that compare two expressions. However, instead of using an equals sign (=), inequalities use symbols like greater than (>), less than (<), greater than or equal to ($\geq$), and less than or equal to ($\leq$). Understanding how to solve inequalities is crucial in various fields, including mathematics, physics, economics, and computer science. This guide will provide a step-by-step approach to solving the given inequality, ensuring clarity and comprehension for readers of all backgrounds.

When tackling inequalities, the primary goal is to isolate the variable, much like in solving equations. The key difference lies in how certain operations affect the inequality sign. Multiplying or dividing both sides of an inequality by a negative number requires flipping the inequality sign. This is a fundamental rule that must be adhered to in order to arrive at the correct solution. We'll explore this concept further as we progress through the solution. Let's break down the process of solving $2(4 + 2x) \geq 5x + 5$ into manageable steps. We will begin by simplifying the inequality using the distributive property, which involves multiplying the term outside the parentheses by each term inside the parentheses. This step is essential for eliminating the parentheses and creating a more workable expression. Next, we will combine like terms, grouping the 'x' terms on one side of the inequality and the constant terms on the other side. This rearrangement helps to isolate the variable and simplifies the inequality further. Once we have combined like terms, we will isolate 'x' by performing the necessary arithmetic operations. This might involve adding or subtracting terms from both sides, or multiplying or dividing both sides by a constant. Remember, if we multiply or divide by a negative number, we must flip the inequality sign. Finally, we will express the solution in its simplest form and represent it graphically on a number line. This visual representation provides a clear understanding of the range of values that satisfy the inequality. By following these steps carefully, we can confidently solve any linear inequality and gain a deeper appreciation for this important mathematical concept.

Step-by-Step Solution of $2(4+2x) \geq 5x + 5$

1. Apply the Distributive Property

Our first step in solving the inequality $2(4 + 2x) \geq 5x + 5$ is to simplify the expression by applying the distributive property. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our case, we need to distribute the '2' across the terms inside the parentheses (4 and 2x). This means we multiply 2 by both 4 and 2x.

So, $2(4 + 2x)$ becomes $(2 * 4) + (2 * 2x)$, which simplifies to $8 + 4x$. Now, substituting this back into the original inequality, we have: $8 + 4x \geq 5x + 5$. Applying the distributive property is a fundamental technique in algebra, allowing us to eliminate parentheses and simplify complex expressions. This step is crucial as it sets the stage for further simplification and ultimately helps us isolate the variable 'x'. Without applying the distributive property, the inequality would remain in a more complex form, making it difficult to proceed with solving it. By breaking down the expression in this way, we make the inequality more manageable and accessible for solving. This process not only simplifies the equation but also provides a clearer understanding of the relationship between the terms involved. In this initial step, we've effectively transformed the inequality into a more solvable form, paving the way for the subsequent steps in the solution process. It is important to perform this step accurately, as any errors in distribution will propagate through the rest of the solution, leading to an incorrect answer. Therefore, careful attention should be paid to the application of the distributive property to ensure the integrity of the solution.

2. Combine Like Terms

After applying the distributive property, our inequality is now $8 + 4x \geq 5x + 5$. The next step involves combining like terms. Combining like terms means grouping terms with the same variable (in this case, 'x') and constant terms on opposite sides of the inequality. Our goal is to isolate 'x' on one side to determine its possible values. To do this, we can subtract $4x$ from both sides of the inequality. This will move the 'x' term from the left side to the right side. Subtracting $4x$ from both sides, we get: $8 + 4x - 4x \geq 5x - 4x + 5$, which simplifies to $8 \geq x + 5$. Now, we need to isolate the 'x' term further by moving the constant term (5) to the left side of the inequality. We can achieve this by subtracting 5 from both sides. Subtracting 5 from both sides, we have: $8 - 5 \geq x + 5 - 5$, which simplifies to $3 \geq x$. This step of combining like terms is crucial because it brings us closer to isolating the variable 'x'. By strategically adding or subtracting terms from both sides of the inequality, we rearrange the expression to make it easier to solve. The key is to perform the same operation on both sides to maintain the balance of the inequality. Combining like terms not only simplifies the inequality but also makes it more intuitive to understand the relationship between 'x' and the constants. It allows us to see the inequality in a clearer, more concise form, which is essential for the final steps of solving for 'x'. In essence, this step is a form of algebraic manipulation that aims to streamline the inequality, making it more amenable to further analysis and solution. By carefully combining like terms, we are effectively organizing the inequality into a form that readily reveals the solution.

3. Isolate the Variable

Following the combination of like terms, our inequality now reads $3 \geq x$. This inequality can also be written as $x \leq 3$. Isolating the variable is the ultimate goal in solving any inequality, and in this case, we have successfully achieved it. The inequality $x \leq 3$ tells us that 'x' can be any value less than or equal to 3. There are no further steps needed to isolate 'x', as it is already on its own on one side of the inequality. The solution $x \leq 3$ represents a range of values that satisfy the original inequality. Any number that is 3 or less will make the inequality $2(4 + 2x) \geq 5x + 5$ true. For example, if we substitute $x = 3$ into the original inequality, we get $2(4 + 2(3)) \geq 5(3) + 5$, which simplifies to $2(10) \geq 20$, or $20 \geq 20$, which is true. Similarly, if we substitute $x = 0$, we get $2(4 + 2(0)) \geq 5(0) + 5$, which simplifies to $8 \geq 5$, also true. On the other hand, if we substitute a value greater than 3, such as $x = 4$, we get $2(4 + 2(4)) \geq 5(4) + 5$, which simplifies to $2(12) \geq 25$, or $24 \geq 25$, which is false. This confirms that the solution $x \leq 3$ is correct. The process of isolating the variable is a fundamental concept in algebra, and it is the culmination of all the previous steps. It is where we finally determine the values that satisfy the given condition. In this instance, the variable 'x' is already isolated, providing a clear and concise solution to the inequality. The simplicity of the final solution underscores the effectiveness of the steps we took to simplify and rearrange the inequality. The solution $x \leq 3$ is not just a numerical answer; it is a statement about the range of possible values that 'x' can take, and this understanding is crucial for applying this solution in various contexts.

Final Answer

Therefore, the solution to the inequality $2(4 + 2x) \geq 5x + 5$ is $x \leq 3$, which corresponds to option C.

The solution $x \leq 3$ to the inequality $2(4 + 2x) \geq 5x + 5$ tells us that any value of 'x' that is less than or equal to 3 will satisfy the inequality. This solution is not just a single number but rather a range of numbers, which is a key characteristic of inequalities. Understanding the implications of this range is crucial for applying this solution in real-world scenarios. For instance, if this inequality represented a constraint in a problem-solving context, we would know that our variable 'x' must fall within this range to be a valid solution. This concept of a range of solutions is fundamental to many areas of mathematics and its applications. In calculus, for example, understanding inequalities is essential for determining intervals of increasing and decreasing functions, as well as for finding maximum and minimum values. In linear programming, inequalities define the feasible region, which represents the set of possible solutions to an optimization problem. Moreover, inequalities play a crucial role in statistics and probability, where they are used to define confidence intervals and hypothesis tests. The ability to interpret and apply the solution of an inequality is a valuable skill that extends beyond the realm of pure mathematics. It is a tool that can be used to model and solve problems in various fields, from economics and finance to engineering and computer science. The solution $x \leq 3$ can be visualized on a number line. To represent this solution graphically, we draw a number line and mark the point 3. Since the inequality includes "equal to," we use a closed circle at 3 to indicate that 3 is part of the solution set. Then, we shade the region to the left of 3, representing all the numbers less than 3. This graphical representation provides a visual understanding of the solution set, making it easier to grasp the concept of a range of values satisfying the inequality. The number line representation also helps to differentiate between strict inequalities (using < or >) and inclusive inequalities (using \leq or \geq). Strict inequalities would be represented with an open circle, indicating that the endpoint is not included in the solution set. In summary, the solution $x \leq 3$ is more than just a final answer; it is a gateway to understanding the broader implications of inequalities in mathematics and beyond. By grasping the concept of a range of solutions and visualizing it graphically, we can develop a deeper appreciation for this important mathematical tool.

When solving inequalities, it's important to be aware of common mistakes that can lead to incorrect solutions. One of the most frequent errors is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Failing to remember this rule is a significant source of error. For example, if we have the inequality $-2x > 4$, we need to divide both sides by -2 to isolate 'x'. When we do this, we must also flip the inequality sign, resulting in $x < -2$. If we forget to flip the sign, we would incorrectly conclude that $x > -2$. To avoid this mistake, it's a good practice to explicitly write down the step where you multiply or divide by a negative number and make a note to flip the inequality sign. Another common mistake is incorrectly applying the distributive property. As we discussed earlier, the distributive property is crucial for simplifying expressions. However, errors can occur if the multiplication is not performed correctly across all terms inside the parentheses. For instance, in the expression $2(4 + 2x)$, if we incorrectly distribute the 2, we might end up with $8 + 2x$ instead of the correct $8 + 4x$. To prevent this, take your time and carefully multiply the term outside the parentheses by each term inside, paying close attention to the signs. A third common mistake is combining like terms incorrectly. This can happen when terms are added or subtracted on the wrong sides of the inequality, or when signs are mishandled. For example, if we have the inequality $8 + 4x \geq 5x + 5$, we need to combine the 'x' terms and the constant terms correctly. If we make an error in this step, such as adding $4x$ to the right side instead of subtracting it, we will arrive at an incorrect inequality. To avoid this, organize your work carefully, writing each step clearly and ensuring that you perform the same operation on both sides of the inequality. Finally, it's essential to check your solution by substituting a value from the solution set back into the original inequality. This will help you verify that your solution is correct. If the value you substitute does not satisfy the original inequality, it indicates that there is an error in your solution, and you need to go back and review your steps. By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in solving inequalities.

In conclusion, solving the inequality $2(4 + 2x) \geq 5x + 5$ involves a series of algebraic steps, including applying the distributive property, combining like terms, and isolating the variable. The correct solution is $x \leq 3$, which means that any value of 'x' less than or equal to 3 will satisfy the inequality. Mastering the process of solving inequalities is a fundamental skill in mathematics, with applications in various fields. By following a systematic approach and being mindful of common mistakes, you can confidently tackle a wide range of inequality problems. The ability to solve inequalities not only strengthens your algebraic skills but also enhances your problem-solving capabilities in real-world contexts. Remember, practice is key to mastering any mathematical concept, so continue to work through examples and challenge yourself with increasingly complex problems. With dedication and perseverance, you can develop a solid understanding of inequalities and their applications.