Solving $-5x + 5 \geq 160 - 10x$ A Step-by-Step Guide

Introduction to Linear Inequalities

In the realm of mathematics, inequalities play a pivotal role in defining relationships between values that are not necessarily equal. Unlike equations that assert equality, inequalities express relationships such as greater than, less than, greater than or equal to, and less than or equal to. Mastering the art of solving inequalities is crucial for various mathematical applications and problem-solving scenarios. In this article, we will delve into a specific linear inequality and dissect the step-by-step process to arrive at the correct solution. Our focus will be on the inequality $-5x + 5 \geq 160 - 10x$, where we aim to determine the range of values for x that satisfy this condition. Understanding how to manipulate and solve such inequalities is not only fundamental in algebra but also essential in real-world applications, where constraints and limitations often need to be expressed mathematically. This exploration will provide a solid foundation for tackling more complex problems involving inequalities. Linear inequalities, characterized by variables raised to the first power, are a cornerstone of algebra, appearing in various mathematical contexts and real-world applications. They are used to describe situations where one quantity is greater than, less than, or equal to another. Understanding how to solve these inequalities is crucial for students and professionals alike.

Understanding the Inequality $-5x + 5 \geq 160 - 10x$

The inequality we're tackling is $-5x + 5 \geq 160 - 10x$. This mathematical statement asserts that the expression on the left side, $-5x + 5$, is greater than or equal to the expression on the right side, $160 - 10x$. Our primary objective is to isolate x on one side of the inequality to determine the range of values that satisfy this condition. To achieve this, we will employ a series of algebraic manipulations, ensuring that each step preserves the integrity of the inequality. This involves strategically adding, subtracting, multiplying, or dividing terms to both sides, all while adhering to the fundamental rules that govern inequality operations. The process requires careful attention to detail, as incorrect manipulations can lead to erroneous solutions. The solution will provide us with a clear understanding of the values of x that make the inequality true, which is a fundamental skill in various mathematical and real-world problem-solving scenarios. Before diving into the solution, it's essential to grasp the core concepts of inequalities and how they differ from equations. While equations seek to find specific values that make both sides equal, inequalities define a range of values that satisfy the relationship. This range can be represented graphically on a number line, providing a visual representation of the solution set.

Step 1: Combining Like Terms Involving 'x'

The initial step in solving the inequality involves gathering all terms containing x on one side of the inequality. This is achieved by adding $10x$ to both sides of the inequality. This operation maintains the balance of the inequality while strategically moving the x terms to one side. The rationale behind this step is to simplify the inequality and make it easier to isolate x. By combining the x terms, we reduce the complexity of the expression, paving the way for subsequent steps. The process mirrors the approach used in solving equations, where like terms are grouped together to streamline the solution process. Performing this addition, we transform the original inequality $-5x + 5 \geq 160 - 10x$ into a more manageable form. This step is a critical juncture in the solution process, setting the stage for the remaining algebraic manipulations. It exemplifies the importance of strategic term rearrangement in solving inequalities and equations alike. After adding $10x$ to both sides, the inequality will look like this: $5x + 5 \geq 160$. This new form is significantly simpler and brings us closer to isolating x. This initial maneuver is a testament to the power of algebraic manipulation in simplifying complex expressions.

Step 2: Isolating the Term with 'x'

Following the combination of x terms, our next objective is to isolate the term containing x on one side of the inequality. In the transformed inequality $5x + 5 \geq 160$, the term $5x$ is accompanied by a constant, $5$. To isolate $5x$, we need to eliminate this constant. This is accomplished by subtracting $5$ from both sides of the inequality. This operation adheres to the principle of maintaining balance in inequalities, ensuring that the relationship between the two sides remains consistent. By subtracting $5$, we effectively remove the constant term from the left side, bringing us closer to isolating x. This step is a crucial element in the process of solving for x, as it simplifies the expression and sets the stage for the final isolation of the variable. The strategic subtraction of constants is a common technique in algebra, employed in both equations and inequalities to streamline the solution process. After performing this subtraction, the inequality takes on a new form, one that is closer to revealing the solution. The transformed inequality will be $5x \geq 155$. This simplified form makes the next step, isolating x, more straightforward and manageable. This meticulous isolation of terms is a hallmark of algebraic problem-solving, highlighting the importance of strategic manipulation.

Step 3: Solving for 'x'

With the term $5x$ isolated on one side of the inequality, the final step in solving for x involves removing the coefficient attached to x. In the inequality $5x \geq 155$, x is multiplied by $5$. To isolate x, we must perform the inverse operation, which is division. We divide both sides of the inequality by $5$, ensuring that the balance of the inequality is maintained. This division effectively cancels out the coefficient, leaving x by itself on the left side. It is a crucial step in determining the range of values that satisfy the inequality. The act of dividing both sides by the coefficient is a fundamental technique in algebra, used to solve for variables in both equations and inequalities. It underscores the importance of inverse operations in mathematical problem-solving. Performing this division will reveal the solution set for x, providing a clear understanding of the values that make the inequality true. After dividing both sides by $5$, we arrive at the solution: $x \geq 31$. This result indicates that any value of x that is greater than or equal to $31$ will satisfy the original inequality. This is the culmination of the step-by-step process, demonstrating the power of algebraic manipulation in solving inequalities.

Interpreting the Solution $x \geq 31$

The solution to the inequality, $x \geq 31$, is a concise mathematical statement that encapsulates a range of values for x. It asserts that x can take on any value that is greater than or equal to $31$. This includes $31$ itself, as well as any number larger than $31$, extending infinitely in the positive direction. Understanding this solution requires grasping the concept of inequalities and their representation on a number line. The solution set can be visualized as a ray on the number line, starting at $31$ and extending to the right, indicating that all values within this range satisfy the inequality. This interpretation is crucial for applying the solution in practical contexts, where x might represent a physical quantity or a constraint in a problem. The inequality $x \geq 31$ provides a clear boundary, defining the permissible values for x. It is a testament to the power of mathematical notation in expressing complex relationships in a succinct and unambiguous manner. The ability to interpret such solutions is a fundamental skill in mathematics, enabling us to apply algebraic results to real-world scenarios.

Verifying the Solution

To ensure the accuracy of our solution, it is crucial to verify that the inequality $x \geq 31$ indeed represents the correct range of values for x. This verification process involves substituting values from the solution set back into the original inequality $-5x + 5 \geq 160 - 10x$ and confirming that the inequality holds true. We can also test values outside the solution set to ensure that they do not satisfy the inequality. This process of verification is a cornerstone of mathematical problem-solving, providing a safeguard against errors and ensuring the reliability of our results. It reinforces the understanding of the inequality and the solution set, solidifying our grasp of the concepts involved. By systematically testing values, we can build confidence in our solution and demonstrate its validity. This step is not merely a formality; it is an integral part of the problem-solving process, ensuring that our mathematical conclusions are sound and accurate. The verification process highlights the importance of rigor in mathematics, where every step is subject to scrutiny and validation.

Testing a Value Within the Solution Set

Let's select a value within the solution set $x \geq 31$, such as $x = 32$, and substitute it into the original inequality $-5x + 5 \geq 160 - 10x$. This substitution will allow us to verify whether the inequality holds true for a value within the solution set. By performing the calculations, we can confirm that the left side of the inequality is indeed greater than or equal to the right side, validating our solution. This process exemplifies the importance of testing in mathematical problem-solving, where substituting values can provide concrete evidence of the correctness of our results. The choice of $x = 32$ is arbitrary; any value greater than or equal to $31$ would serve the same purpose. The key is to select a representative value that falls within the solution set. The substitution process is a powerful tool for understanding the behavior of inequalities and confirming the validity of our algebraic manipulations. It reinforces the connection between the algebraic solution and the numerical values that satisfy the inequality. After substituting $x = 32$, the inequality becomes $-5(32) + 5 \geq 160 - 10(32)$, which simplifies to $-155 \geq -160$, a true statement. This confirms that our solution is valid for at least one value within the solution set.

Testing a Value Outside the Solution Set

Now, let's select a value outside the solution set $x \geq 31$, such as $x = 30$, and substitute it into the original inequality $-5x + 5 \geq 160 - 10x$. This substitution will allow us to verify whether the inequality does not hold true for a value outside the solution set, further validating our solution. By performing the calculations, we can confirm that the left side of the inequality is not greater than or equal to the right side, demonstrating that our solution accurately defines the boundary between values that satisfy the inequality and those that do not. This process provides a crucial check, ensuring that our solution is not only inclusive of the correct values but also exclusive of incorrect ones. The choice of $x = 30$ is strategic, as it is a value immediately adjacent to the boundary of the solution set. This choice helps to confirm the precision of our solution. The substitution process reinforces the understanding of inequalities and their solutions as defining a specific range of values. After substituting $x = 30$, the inequality becomes $-5(30) + 5 \geq 160 - 10(30)$, which simplifies to $-145 \geq -140$, a false statement. This confirms that our solution accurately excludes values outside the solution set.

Conclusion

In conclusion, the correct inequality that represents all the solutions of $-5x + 5 \geq 160 - 10x$ is $x \geq 31$, which corresponds to option C. This solution was derived through a step-by-step process of algebraic manipulation, involving the combination of like terms, the isolation of the term with x, and the final division to solve for x. The solution was further verified by substituting values within and outside the solution set, confirming its accuracy and validity. This comprehensive approach underscores the importance of a systematic methodology in solving inequalities and the need for rigorous verification to ensure the correctness of mathematical results. The ability to solve linear inequalities is a fundamental skill in mathematics, with applications spanning various fields and disciplines. Mastering this skill requires a solid understanding of algebraic principles and a commitment to careful and methodical problem-solving. The journey through this specific inequality serves as a testament to the power of algebraic techniques and the importance of precision in mathematical reasoning. The solution not only provides the answer to the specific problem but also reinforces the broader concepts of inequalities and their applications in mathematics and beyond.