Solving The Inequality $1 > -5 + 6w$ A Step-by-Step Guide

Inequalities are a fundamental concept in mathematics, representing relationships where one value is not necessarily equal to another. Unlike equations that seek to find specific solutions, inequalities define a range of possible values. Mastering the techniques to solve inequalities is crucial for various mathematical applications, including calculus, optimization problems, and real-world modeling. This comprehensive guide will walk you through the process of solving the inequality $1 > -5 + 6w$, providing a step-by-step explanation to ensure clarity and understanding. We will explore the underlying principles, the algebraic manipulations involved, and the interpretation of the final solution. Understanding inequalities is not just about finding the answer; it's about grasping the concept of a range of values that satisfy a given condition. Let's dive into the process of solving this inequality, building a solid foundation for more advanced mathematical concepts. This guide aims to provide a clear and concise explanation, making it accessible to learners of all levels. By the end of this guide, you will be equipped with the skills and knowledge to tackle similar inequality problems with confidence.

Understanding Inequalities

Before we delve into the specific problem, let's establish a firm grasp of what inequalities represent. Inequalities, unlike equations, compare two expressions that are not necessarily equal. They use symbols like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to) to express these relationships. The inequality $1 > -5 + 6w$ states that the value of 1 is strictly greater than the expression $-5 + 6w$. Our goal is to isolate the variable w and determine the range of values that satisfy this condition. This process involves applying algebraic operations to both sides of the inequality, similar to solving equations, but with a crucial difference: multiplying or dividing by a negative number reverses the direction of the inequality sign. Understanding this rule is paramount to arriving at the correct solution. Visualizing inequalities on a number line can also be beneficial. For instance, the solution to our inequality will represent a set of numbers that lie to the left or right of a specific point, depending on the direction of the inequality. This visual representation helps to solidify the concept and makes it easier to interpret the solution in a real-world context. In essence, inequalities provide a powerful tool for modeling situations where exact values are not known, but a range of possibilities can be defined. This makes them indispensable in fields such as economics, engineering, and computer science.

Step-by-Step Solution

Now, let's proceed with solving the inequality $1 > -5 + 6w$ step by step. The primary objective is to isolate the variable w on one side of the inequality. To achieve this, we will employ a series of algebraic manipulations while adhering to the rules governing inequalities. The first step involves eliminating the constant term (-5) from the right side of the inequality. We accomplish this by adding 5 to both sides of the inequality. This maintains the balance of the inequality while simplifying the expression. Adding 5 to both sides gives us $1 + 5 > -5 + 6w + 5$, which simplifies to $6 > 6w$. Next, we need to isolate w by eliminating the coefficient 6. This is achieved by dividing both sides of the inequality by 6. Since we are dividing by a positive number, the direction of the inequality sign remains unchanged. Dividing both sides by 6 yields $6/6 > 6w/6$, which simplifies to $1 > w$. This result indicates that w must be less than 1 for the inequality to hold true. It's important to note that if we had divided by a negative number, we would have needed to reverse the inequality sign. The final solution, $1 > w$, can also be written as $w < 1$. Both expressions are equivalent and convey the same information: the set of all values of w that are less than 1. This step-by-step approach ensures clarity and minimizes the chances of making errors. Each step is justified by the properties of inequalities, leading us to the correct solution.

Detailed Solution and Explanation

Let's break down the solution process with a more detailed explanation of each step. Our initial inequality is $1 > -5 + 6w$. The first goal is to isolate the term containing the variable w. This involves removing any constants that are added or subtracted from this term. In our case, we have -5 added to $6w$. To eliminate -5, we perform the inverse operation, which is adding 5. We add 5 to both sides of the inequality to maintain balance. This yields: $1 + 5 > -5 + 6w + 5$. Simplifying both sides, we get $6 > 6w$. Now, we have the term $6w$ on the right side. To isolate w, we need to eliminate the coefficient 6. This is done by dividing both sides of the inequality by 6. It's crucial to remember that when we divide (or multiply) an inequality by a negative number, we must reverse the direction of the inequality sign. However, in this case, we are dividing by a positive number (6), so the inequality sign remains the same. Dividing both sides by 6 gives us: $6/6 > 6w/6$. Simplifying, we obtain $1 > w$. This is our solution, which states that 1 is greater than w. Alternatively, we can express this as $w < 1$, which reads as w is less than 1. This solution represents an infinite set of values for w. Any number less than 1 will satisfy the original inequality. For example, 0, -1, -2, and so on, are all valid solutions. To visualize this solution, we can represent it on a number line. We would draw an open circle at 1 (since w is strictly less than 1, not less than or equal to 1) and shade the region to the left, indicating all values less than 1. This detailed explanation clarifies the reasoning behind each step, ensuring a thorough understanding of the solution process.

Expressing the Solution

The solution to the inequality $1 > -5 + 6w$ is $w < 1$. However, there are different ways to express this solution, each providing a slightly different perspective. We've already seen the algebraic representation, $w < 1$, which is concise and commonly used. This notation directly states that w can take any value less than 1. Another way to express the solution is using interval notation. Interval notation is a convenient way to represent a range of values on the number line. For the inequality $w < 1$, the interval notation is $(-\infty, 1)$. The parenthesis indicates that the endpoint (1) is not included in the solution set, which aligns with the strict inequality $w < 1$. The $-\infty$ symbol represents negative infinity, indicating that the solution extends indefinitely in the negative direction. Visualizing the solution on a number line provides yet another way to express it. We would draw a number line, mark the point 1, and draw an open circle at 1 to signify that it is not included in the solution. Then, we would shade the portion of the number line to the left of 1, representing all values less than 1. This visual representation is particularly helpful for understanding the concept of a range of solutions. Finally, we can describe the solution in words. We can say that the solution is the set of all real numbers less than 1. This verbal description provides a clear and intuitive understanding of the solution set. Each of these methods – algebraic notation, interval notation, number line representation, and verbal description – offers a valuable way to express and interpret the solution to the inequality. Choosing the most appropriate method depends on the context and the desired level of clarity.

Verification and Practical Examples

After solving an inequality, it's crucial to verify the solution to ensure accuracy. This involves substituting values from the solution set back into the original inequality and checking if the inequality holds true. For our inequality, $1 > -5 + 6w$, the solution is $w < 1$. To verify this, let's choose a value less than 1, say $w = 0$. Substituting $w = 0$ into the original inequality, we get: $1 > -5 + 6(0)$, which simplifies to $1 > -5$. This is a true statement, confirming that $w = 0$ is indeed a solution. Now, let's choose another value less than 1, such as $w = -1$. Substituting $w = -1$, we get: $1 > -5 + 6(-1)$, which simplifies to $1 > -11$. Again, this is a true statement, further validating our solution. It's also important to test a value that is not in the solution set. Let's choose $w = 2$, which is greater than 1. Substituting $w = 2$, we get: $1 > -5 + 6(2)$, which simplifies to $1 > 7$. This is a false statement, confirming that values greater than 1 do not satisfy the inequality. This verification process provides strong evidence that our solution, $w < 1$, is correct. Inequalities have numerous practical applications in real-world scenarios. For instance, they can be used to model constraints in optimization problems, determine the range of values for a variable in a physical system, or analyze financial situations. Understanding how to solve and interpret inequalities is therefore an essential skill in various fields.

Common Mistakes to Avoid

When solving inequalities, several common mistakes can lead to incorrect solutions. Being aware of these pitfalls can help you avoid them and ensure accuracy. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. This rule is crucial and must be applied consistently. For example, if we had an inequality like $-2w < 4$, dividing both sides by -2 would require flipping the inequality sign, resulting in $w > -2$. Another common mistake is incorrectly applying the order of operations. It's essential to follow the correct order (PEMDAS/BODMAS) when simplifying expressions on both sides of the inequality. This includes dealing with parentheses, exponents, multiplication and division, and addition and subtraction in the correct sequence. Failing to distribute properly can also lead to errors. If an expression like $3(w + 2)$ appears in the inequality, it's necessary to distribute the 3 to both terms inside the parentheses, resulting in $3w + 6$. Another pitfall is making errors when combining like terms. Ensure that you are only combining terms that have the same variable and exponent. For instance, $2w + 3w$ can be combined to give $5w$, but $2w + 3w^2$ cannot be combined. Finally, it's essential to remember that the solution to an inequality is often a range of values, not just a single value. Expressing the solution correctly using interval notation or a number line is crucial for conveying the complete solution set. By being mindful of these common mistakes and practicing regularly, you can improve your accuracy and confidence in solving inequalities.

Conclusion

In conclusion, solving inequalities is a fundamental skill in mathematics with wide-ranging applications. This guide has provided a comprehensive step-by-step approach to solving the inequality $1 > -5 + 6w$, emphasizing the underlying principles and techniques involved. We began by understanding the concept of inequalities and how they differ from equations. We then meticulously worked through the solution process, isolating the variable w using algebraic manipulations. Each step was explained in detail, highlighting the importance of maintaining the balance of the inequality and remembering to reverse the inequality sign when multiplying or dividing by a negative number. The solution, $w < 1$, represents the set of all values less than 1 that satisfy the original inequality. We explored different ways to express this solution, including algebraic notation, interval notation, number line representation, and verbal description. Verification of the solution was emphasized as a crucial step to ensure accuracy. By substituting values from the solution set back into the original inequality, we confirmed that our solution was correct. Practical examples were discussed to illustrate the real-world applications of inequalities in various fields. Finally, we addressed common mistakes to avoid when solving inequalities, such as forgetting to reverse the inequality sign or incorrectly applying the order of operations. By mastering the concepts and techniques presented in this guide, you will be well-equipped to solve a wide range of inequalities with confidence and accuracy. Inequalities are not just abstract mathematical concepts; they are powerful tools for modeling and solving problems in the real world. Continuing to practice and apply these skills will further solidify your understanding and enhance your problem-solving abilities.