Solving Inequalities A Step By Step Guide To \( \frac{4}{3}w + 2 \leq -w - \frac{2}{7} \)

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Introduction

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations, which assert the equality of two expressions, inequalities express a range of possibilities. This article delves into the step-by-step process of solving the inequality 43w+2≤−w−27{ \frac{4}{3}w + 2 \leq -w - \frac{2}{7} }, providing a comprehensive guide for students and enthusiasts alike. Mastering the techniques for solving inequalities is fundamental for various mathematical applications, including optimization problems, calculus, and real-world scenarios involving constraints and limitations.

Understanding inequalities is also vital for grasping concepts in advanced mathematics. For instance, in calculus, inequalities are used to define intervals of convergence for series and to determine the behavior of functions. In linear programming, inequalities represent constraints that define the feasible region for optimization problems. Furthermore, in fields like economics and engineering, inequalities are used to model constraints such as budget limitations, resource availability, and performance requirements. Thus, a solid foundation in solving inequalities is essential for anyone pursuing studies or careers in these areas.

The given inequality, 43w+2≤−w−27{ \frac{4}{3}w + 2 \leq -w - \frac{2}{7} }, involves rational coefficients and constant terms, making it a typical example encountered in algebra. Solving this inequality requires a combination of algebraic manipulations, including adding and subtracting terms, multiplying by a constant, and simplifying fractions. Each step must be performed carefully to maintain the integrity of the inequality. The solution will be a range of values for w{ w } that satisfy the given condition, which can be represented graphically on a number line or expressed in interval notation.

This article aims to provide a clear and detailed explanation of each step involved in solving the inequality. By breaking down the process into manageable parts and offering insights into the underlying principles, we hope to empower readers with the skills and confidence to tackle similar problems. Whether you are a student preparing for an exam or simply a mathematics enthusiast seeking to expand your knowledge, this guide will serve as a valuable resource. So, let's embark on this journey to unravel the intricacies of solving inequalities and discover the power of mathematical reasoning.

Step 1: Clearing Fractions

The first crucial step in solving the inequality 43w+2≤−w−27{ \frac{4}{3}w + 2 \leq -w - \frac{2}{7} } involves eliminating the fractions. Fractions can often complicate algebraic manipulations, so clearing them simplifies the process significantly. To achieve this, we need to find the least common multiple (LCM) of the denominators present in the inequality. In this case, the denominators are 3 and 7. The LCM of 3 and 7 is 21, as it is the smallest number divisible by both 3 and 7. Identifying the LCM is a fundamental skill in arithmetic and is essential for operations involving fractions.

Once we have determined the LCM, we multiply every term in the inequality by this value. This ensures that the fractions are canceled out, leaving us with an inequality involving only integer coefficients. Multiplying each term by the LCM is a valid algebraic operation because it maintains the balance of the inequality, provided that we multiply every term on both sides. This is a direct application of the multiplication property of inequality, which states that multiplying both sides of an inequality by a positive number does not change the direction of the inequality.

Applying this to our inequality, we multiply each term by 21:

21(43w+2)≤21(−w−27){ 21 \left( \frac{4}{3}w + 2 \right) \leq 21 \left( -w - \frac{2}{7} \right) }

Now, we distribute the 21 to each term inside the parentheses:

21⋅43w+21⋅2≤21⋅(−w)−21⋅27{ 21 \cdot \frac{4}{3}w + 21 \cdot 2 \leq 21 \cdot (-w) - 21 \cdot \frac{2}{7} }

Next, we perform the multiplication and simplify each term. For the first term, 21⋅43w{ 21 \cdot \frac{4}{3}w }, we can simplify the fraction by dividing 21 by 3, which gives us 7. Then, we multiply 7 by 4 to get 28. Thus, the first term simplifies to 28w. For the second term, 21⋅2{ 21 \cdot 2 }, the multiplication is straightforward, resulting in 42. On the right side of the inequality, the first term 21⋅(−w){ 21 \cdot (-w) } is simply -21w. For the last term, −21⋅27{ -21 \cdot \frac{2}{7} }, we divide 21 by 7, which gives us 3. Then, we multiply 3 by 2 to get 6, so the term simplifies to -6. Putting it all together, we have:

28w+42≤−21w−6{ 28w + 42 \leq -21w - 6 }

This resulting inequality is now free of fractions, making it much easier to manipulate and solve. The process of clearing fractions is a common technique in algebra, not only for inequalities but also for equations involving fractions. By eliminating the fractions, we transform the inequality into a more manageable form, setting the stage for the next steps in the solution process. This step demonstrates the importance of simplifying expressions before proceeding with more complex operations, a principle that applies across various areas of mathematics.

Step 2: Isolating the Variable

After successfully clearing the fractions, the next critical step in solving the inequality 28w+42≤−21w−6{ 28w + 42 \leq -21w - 6 } is to isolate the variable w{ w }. This involves rearranging the terms in the inequality so that all terms containing w{ w } are on one side and all constant terms are on the other side. The goal is to get the variable w{ w } by itself, which will reveal the range of values that satisfy the inequality. Isolating the variable is a fundamental technique in algebra and is essential for solving equations and inequalities of various types.

To begin isolating w{ w }, we first need to move all terms containing w{ w } to one side of the inequality. In this case, it's often convenient to move the terms to the side where the coefficient of w{ w } is larger, which helps avoid dealing with negative coefficients. Looking at our inequality, we have 28w{ 28w } on the left side and −21w{ -21w } on the right side. Since 28 is greater than -21, we will move the −21w{ -21w } term to the left side. To do this, we add 21w{ 21w } to both sides of the inequality. This operation maintains the balance of the inequality, as adding the same quantity to both sides does not change the relationship between them. This is an application of the addition property of inequality.

Adding 21w{ 21w } to both sides, we get:

28w+21w+42≤−21w+21w−6{ 28w + 21w + 42 \leq -21w + 21w - 6 }

Simplifying, we combine the like terms on each side:

49w+42≤−6{ 49w + 42 \leq -6 }

Now that we have all the w{ w } terms on the left side, we need to move all the constant terms to the right side. We have the constant term 42 on the left side, which we need to move to the right side. To do this, we subtract 42 from both sides of the inequality. Again, this operation maintains the balance of the inequality, as subtracting the same quantity from both sides does not change the relationship between them. This is an application of the subtraction property of inequality.

Subtracting 42 from both sides, we get:

49w+42−42≤−6−42{ 49w + 42 - 42 \leq -6 - 42 }

Simplifying, we have:

49w≤−48{ 49w \leq -48 }

At this point, we have successfully isolated the variable term 49w{ 49w } on the left side and the constant term −48{ -48 } on the right side. The inequality is now in a simplified form that allows us to solve for w{ w } directly. This process of isolating the variable is a crucial step in solving any algebraic equation or inequality. By systematically moving terms and simplifying, we bring the inequality closer to its solution.

Step 3: Solving for w

Having successfully isolated the variable term, we are now at the final step of solving the inequality 49w≤−48{ 49w \leq -48 }. The objective here is to determine the range of values for w{ w } that satisfy this inequality. To do this, we need to get w{ w } completely by itself on one side of the inequality. This involves dividing both sides of the inequality by the coefficient of w{ w }, which in this case is 49. Division is a fundamental arithmetic operation, and when applied to inequalities, it requires careful consideration of the sign of the divisor.

The division property of inequality states that if we divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, if we divide by a negative number, the direction of the inequality must be reversed. In our case, we are dividing by 49, which is a positive number, so the direction of the inequality will not change. This is a crucial point to remember, as neglecting to reverse the inequality sign when dividing by a negative number is a common mistake that leads to an incorrect solution.

Dividing both sides of the inequality by 49, we get:

49w49≤−4849{ \frac{49w}{49} \leq \frac{-48}{49} }

Simplifying the left side, 49w49{ \frac{49w}{49} } reduces to w{ w }, as 49 divided by 49 is 1. The right side, −4849{ \frac{-48}{49} }, is a fraction that cannot be simplified further, so we leave it as it is. Thus, we have:

w≤−4849{ w \leq -\frac{48}{49} }

This is the solution to the inequality. It tells us that w{ w } can be any value that is less than or equal to −4849{ -\frac{48}{49} }. This range of values represents all the possible solutions to the original inequality. The solution can be expressed in various forms, including inequality notation, interval notation, and graphically on a number line. Each representation provides a different perspective on the solution set.

In inequality notation, as we have it now, the solution is w≤−4849{ w \leq -\frac{48}{49} }. This notation directly states the condition that w{ w } must satisfy.

In interval notation, the solution is represented as (−∞,−4849]{ (-\infty, -\frac{48}{49}] }. This notation uses parentheses and brackets to indicate the range of values. The parenthesis ({ ( } indicates that the endpoint is not included in the interval, while the bracket [{ [ } indicates that the endpoint is included. In this case, −∞{ -\infty } is always represented with a parenthesis because infinity is not a specific number but rather a concept of unboundedness. The bracket around −4849{ -\frac{48}{49} } indicates that this value is included in the solution set.

Graphically, the solution can be represented on a number line. We draw a number line and mark the point −4849{ -\frac{48}{49} }. Since the inequality includes “equal to,” we use a closed circle (or a filled-in dot) at this point to indicate that it is included in the solution. Then, we shade the region to the left of this point, representing all values less than −4849{ -\frac{48}{49} }. This shaded region visually represents the solution set.

Therefore, the solution to the inequality 43w+2≤−w−27{ \frac{4}{3}w + 2 \leq -w - \frac{2}{7} } is w≤−4849{ w \leq -\frac{48}{49} }, which can be expressed in interval notation as (−∞,−4849]{ (-\infty, -\frac{48}{49}] } and graphically represented on a number line with a closed circle at −4849{ -\frac{48}{49} } and shading to the left. This comprehensive solution demonstrates the power of algebraic manipulation and the importance of understanding the properties of inequalities.

Conclusion

In this comprehensive guide, we have meticulously walked through the process of solving the inequality 43w+2≤−w−27{ \frac{4}{3}w + 2 \leq -w - \frac{2}{7} }. Each step, from clearing fractions to isolating the variable and finally solving for w{ w }, has been explained in detail to provide a clear understanding of the underlying principles and techniques. By breaking down the problem into manageable parts, we have demonstrated how complex inequalities can be tackled with systematic algebraic manipulations. The solution, w≤−4849{ w \leq -\frac{48}{49} }, represents a range of values that satisfy the given condition, and we have explored how to express this solution in inequality notation, interval notation, and graphically on a number line.

Throughout this process, we emphasized the importance of maintaining the balance of the inequality by performing the same operations on both sides. This is a fundamental concept in algebra and is crucial for solving any equation or inequality. We also highlighted the significance of the division property of inequality, which requires reversing the inequality sign when dividing by a negative number. Avoiding this common mistake is essential for arriving at the correct solution. The techniques and strategies discussed here are not only applicable to this specific inequality but also serve as a foundation for solving a wide range of mathematical problems.

The ability to solve inequalities is a valuable skill in mathematics and has practical applications in various fields. Inequalities are used to model constraints and limitations in real-world scenarios, such as budget constraints in economics, resource limitations in engineering, and performance requirements in computer science. Understanding how to solve inequalities allows us to make informed decisions and optimize outcomes in these situations. Furthermore, inequalities play a crucial role in advanced mathematical concepts, such as calculus, where they are used to define intervals of convergence and to analyze the behavior of functions.

By mastering the techniques presented in this guide, readers will be well-equipped to tackle similar inequalities and to apply these skills in more complex mathematical contexts. Whether you are a student preparing for an exam or a professional working on a real-world problem, the ability to solve inequalities is an invaluable asset. We encourage you to practice these techniques with various examples to solidify your understanding and to develop confidence in your problem-solving abilities. The journey through mathematics is one of continuous learning and discovery, and mastering inequalities is a significant step towards unlocking the power of mathematical reasoning.

In conclusion, solving the inequality 43w+2≤−w−27{ \frac{4}{3}w + 2 \leq -w - \frac{2}{7} } is a testament to the power of algebraic manipulation and the importance of understanding fundamental mathematical principles. The solution, w≤−4849{ w \leq -\frac{48}{49} }, is not just a numerical answer but a representation of a range of possibilities. By following the step-by-step process outlined in this guide, we hope to have demystified the process and empowered readers to confidently tackle similar challenges in the future.