Mastering Inequalities: A Step-by-Step Guide

Understanding the Inequality: $29 - 2(3 - 5w) ">=" 13$

When we talk about solving inequalities, we're essentially looking for a range of values for a variable that make a given statement true. Unlike equations where we often find a single specific value, inequalities usually result in a set of solutions. The inequality we'll be diving into today is $29 - 2(3 - 5w) ">=" 13$. This might look a little intimidating at first glance with the parentheses and the greater than or equal to sign, but fear not! We're going to break it down into manageable steps. The core idea is to isolate the variable, in this case, 'w', on one side of the inequality sign. Think of it like a balancing scale; whatever operation you perform on one side, you must also perform on the other to maintain the truth of the statement. We'll start by simplifying the expression, then systematically move terms around until 'w' is by itself. Remember, the rules for manipulating inequalities are very similar to those for equations, with one crucial difference: if you multiply or divide both sides by a negative number, you must flip the inequality sign. We'll keep this in mind as we progress. This particular inequality involves distribution, subtraction, and ultimately, division. Each step is designed to peel away the layers surrounding 'w' so we can clearly see its possible values. We're aiming to find all the numbers 'w' can be to ensure that $29 - 2(3 - 5w)$ is always greater than or equal to 13. It’s a fundamental concept in algebra that opens the door to understanding more complex mathematical relationships and problem-solving scenarios. So, let's roll up our sleeves and get ready to conquer this inequality, transforming it from a complex expression into a clear solution.

Step 1: Distribute and Simplify

The first hurdle in solving $29 - 2(3 - 5w) ">=" 13$ is dealing with the parentheses. This is where the distributive property comes into play. Remember, the distributive property states that $a(b + c) = ab + ac$. In our case, we have $-2$ multiplying the expression inside the parentheses, $(3 - 5w)$. So, we need to multiply $-2$ by both $3$ and $-5w$. Let's do that: $-2 \times 3 = -6$ and $-2 \times (-5w) = +10w$. Now, we can rewrite the inequality with the distributed terms: $29 - 6 + 10w ">=" 13$. After distributing, the next logical step is to combine any like terms on the left side of the inequality. We have the constants $29$ and $-6$. Combining these gives us $29 - 6 = 23$. So, the inequality simplifies to $23 + 10w ">=" 13$. This step is crucial because it reduces the complexity of the inequality, making it easier to work with. We've successfully eliminated the parentheses and combined the constant terms, bringing us closer to isolating our variable 'w'. It's like clearing the clutter from a desk before you start a detailed task. Each simplification makes the path forward clearer and less prone to errors. Think about it: if we hadn't distributed properly, every subsequent step would be based on a faulty foundation. This initial phase of distribution and simplification is paramount to arriving at the correct solution. It ensures that we are working with the most reduced and accurate form of the original inequality, setting the stage for the subsequent algebraic manipulations that will reveal the value of 'w'. This careful attention to the order of operations and the rules of distribution is a hallmark of successful algebraic problem-solving, and it’s a skill that will serve you well across many mathematical challenges.

Step 2: Isolate the Variable Term

Now that we have the simplified inequality $23 + 10w ">=" 13$, our next goal is to get the term containing our variable, $10w$, all by itself on one side. Currently, it's being added to $23$. To undo the addition of $23$, we need to perform the inverse operation, which is subtraction. So, we will subtract $23$ from both sides of the inequality. This maintains the balance of the inequality. On the left side, we have $23 + 10w - 23$. The $23$ and $-23$ cancel each other out, leaving us with just $10w$. On the right side, we perform the same operation: $13 - 23$. Calculating this gives us $-10$. Therefore, the inequality now becomes $10w ">=" -10$. This step is vital because it isolates the term with the variable, bringing us one step closer to finding the actual value of 'w'. We've systematically removed the constant term from the side with the variable, which is a standard strategy in solving algebraic equations and inequalities. It’s about peeling back the layers, one operation at a time. By subtracting $23$ from both sides, we ensure that the inequality remains true. If we only subtracted $23$ from the left side, the relationship between the two sides would be broken. The principle of performing the same operation on both sides is a cornerstone of algebraic manipulation. This stage is about strategic simplification, moving towards the final isolation of 'w'. It requires careful arithmetic and a clear understanding of inverse operations. As we move forward, remember that the goal is to achieve a state where 'w' is the subject, and all other numbers and operations are on the opposing side. This isolation technique is fundamental and applicable to a vast array of algebraic problems, making it an indispensable skill in your mathematical toolkit.

Step 3: Solve for 'w'

We've reached the final stage in solving $10w ">=" -10$. The variable term $10w$ is now isolated. To find the value of 'w', we need to undo the multiplication by $10$. The inverse operation of multiplication is division. So, we will divide both sides of the inequality by $10$. Remember, we are dividing by a positive number ($10$), so the inequality sign does not change. On the left side, $10w / 10$ simplifies to just $w$. On the right side, we have $-10 / 10$. Dividing $-10$ by $10$ gives us $-1$. Therefore, the solution to the inequality is $w ">=" -1$. This means that any value of 'w' that is greater than or equal to $-1$ will satisfy the original inequality $29 - 2(3 - 5w) ">=" 13$. For example, if we pick $w = 0$, which is greater than $-1$, let's check: $29 - 2(3 - 5(0)) = 29 - 2(3) = 29 - 6 = 23$. And $23 ">=" 13$, which is true. If we pick $w = -1$, we get $29 - 2(3 - 5(-1)) = 29 - 2(3 + 5) = 29 - 2(8) = 29 - 16 = 13$. And $13 ">=" 13$, which is also true. If we pick a value less than $-1$, say $w = -2$: $29 - 2(3 - 5(-2)) = 29 - 2(3 + 10) = 29 - 2(13) = 29 - 26 = 3$. And $3 ">=" 13$, which is false. This confirms our solution. The process of dividing to isolate the variable is the final step in unraveling the inequality. It's a straightforward calculation once the variable term is alone. Understanding when to flip the inequality sign is a critical detail that distinguishes inequality solving from equation solving. In this instance, dividing by a positive number meant the sign remained the same. This final result, $w ">=" -1$, represents the complete set of solutions. It's a powerful outcome that encapsulates infinite possibilities for 'w', all bound by this simple condition. Mastering this process is key to progressing in algebra and beyond.

Conclusion: The Power of Inequalities

We've successfully navigated the journey of solving the inequality $29 - 2(3 - 5w) ">=" 13$, arriving at the solution $w ">=" -1$. This might seem like a simple algebraic exercise, but it embodies a fundamental concept with wide-reaching applications. Understanding inequalities is not just about finding a range of numbers; it's about grasping the concept of relationships between quantities that aren't necessarily equal. Whether you're dealing with budgeting, physics, engineering, or even everyday decision-making, inequalities are constantly at play. They allow us to define boundaries, set limits, and express conditions. For instance, a speed limit is an inequality (speed $"<=" 65$ mph), or a minimum age requirement for a service is another (age $">=" 18$). The ability to manipulate and solve these expressions empowers you to analyze situations, make informed predictions, and set parameters in a quantitative way. This step-by-step process—distribution, simplification, isolation, and final solution—is a blueprint applicable to a vast array of algebraic problems. Remember the golden rule: when multiplying or dividing by a negative number, flip that inequality sign! It’s a small detail that makes a monumental difference. Practice is key to solidifying these skills. The more inequalities you solve, the more intuitive the process will become. You'll start to recognize patterns and anticipate steps, making you a more confident and capable mathematician. Don't shy away from tackling different types of inequalities; each one presents a unique opportunity to learn and grow. The world of mathematics is built on these foundational concepts, and a firm grasp of inequalities will undoubtedly serve you well in your academic pursuits and beyond. For further exploration into the fascinating world of algebra and inequalities, I highly recommend checking out resources from Khan Academy for excellent tutorials and practice problems, or exploring the extensive mathematical knowledge available on Wikipedia's Algebra page.