Solving $2 leqslant \frac{2}{3}(5-x)$ A Step-by-Step Guide

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Introduction to Inequalities in Mathematics

In the realm of mathematics, inequalities play a pivotal role in describing relationships where quantities are not necessarily equal. Unlike equations that assert the equality between two expressions, inequalities express a range of possible values. Understanding and solving inequalities is a fundamental skill in algebra and calculus, with applications spanning various fields, including economics, physics, and computer science. This article delves into the process of solving a specific inequality: 2 ≤ (2/3)(5-x), providing a step-by-step guide to help you master this essential mathematical concept. We will explore the properties of inequalities, the techniques for isolating the variable, and the interpretation of the solution set. Whether you are a student grappling with algebraic concepts or a seasoned mathematician seeking a refresher, this comprehensive guide will equip you with the knowledge and skills to confidently tackle inequalities.

Understanding the Basics of Inequalities

Before diving into the specific problem, let's establish a solid foundation by understanding the basics of inequalities. Inequalities use symbols to compare two values: '<' (less than), '>' (greater than), '≤' (less than or equal to), and '≥' (greater than or equal to). The inequality 2 ≤ (2/3)(5-x) involves the '≤' symbol, indicating that the expression on the left (2) is less than or equal to the expression on the right ((2/3)(5-x)). To solve an inequality, our goal is to isolate the variable (x in this case) on one side of the inequality symbol, just as we do with equations. However, there's a crucial difference: multiplying or dividing an inequality by a negative number reverses the direction of the inequality symbol. This is a critical rule to remember and will be discussed in detail later. Inequalities often have a range of solutions, rather than a single solution like equations. This range is typically represented as an interval on a number line or using interval notation. For example, x > 3 means x can be any value greater than 3, and this is represented as (3, ∞) in interval notation. The ability to manipulate and solve inequalities is crucial for various mathematical applications, including optimization problems, constraint satisfaction, and the analysis of functions and their behavior. Understanding the nuances of inequalities empowers us to model real-world scenarios involving constraints and ranges of values, making it an indispensable tool in problem-solving across diverse disciplines.

Step-by-Step Solution to 2 ≤ (2/3)(5-x)

Now, let's tackle the inequality 2 ≤ (2/3)(5-x) step-by-step, providing a clear and concise explanation for each operation. This will not only lead us to the solution but also reinforce the fundamental principles of solving inequalities. Our aim is to isolate 'x' on one side of the inequality, revealing the range of values that satisfy the given condition.

Step 1: Distribute the Constant

The first step in solving this inequality is to simplify the right-hand side by distributing the constant (2/3) across the terms inside the parentheses. This means multiplying (2/3) by both 5 and -x. Performing this distribution, we get:

2 ≤ (2/3) * 5 - (2/3) * x

This simplifies to:

2 ≤ 10/3 - (2/3)x

Distributing the constant helps us eliminate the parentheses and brings us closer to isolating the variable 'x'. This is a standard algebraic technique that applies to both equations and inequalities. By correctly applying the distributive property, we ensure that the inequality remains balanced and the subsequent steps will lead to an accurate solution.

Step 2: Eliminate the Fraction

To make the inequality easier to work with, the next logical step is to eliminate the fraction. We can achieve this by multiplying both sides of the inequality by the denominator, which in this case is 3. Multiplying both sides by 3 maintains the balance of the inequality and clears the fractions. This is a crucial step in simplifying inequalities, particularly when dealing with rational expressions. When multiplying inequalities, we must be mindful of the sign of the number we are multiplying by. Since we are multiplying by a positive number (3), the direction of the inequality symbol remains unchanged. However, if we were to multiply by a negative number, we would need to reverse the inequality sign.

Multiplying both sides by 3, we get:

3 * 2 ≤ 3 * (10/3 - (2/3)x)

This simplifies to:

6 ≤ 10 - 2x

Now, the inequality is free of fractions, making it more manageable for further manipulation. This simplification is a common strategy in solving algebraic problems, allowing us to focus on isolating the variable without the added complexity of fractional coefficients.

Step 3: Isolate the Term with 'x'

Our next goal is to isolate the term containing 'x' on one side of the inequality. In this case, the term is -2x. To isolate it, we need to eliminate the constant term (10) from the right side. We can do this by subtracting 10 from both sides of the inequality. Subtracting the same value from both sides maintains the balance of the inequality and moves us closer to isolating 'x'.

Subtracting 10 from both sides, we get:

6 - 10 ≤ 10 - 2x - 10

This simplifies to:

-4 ≤ -2x

Now, the term with 'x' is isolated on the right side, and we are one step closer to finding the solution. Isolating the variable term is a fundamental technique in solving inequalities and equations, allowing us to focus on the final step of isolating the variable itself.

Step 4: Solve for 'x'

Finally, to solve for 'x', we need to isolate 'x' completely. This means getting rid of the coefficient -2 that is multiplying 'x'. To do this, we divide both sides of the inequality by -2. However, there's a crucial rule we must remember: when we divide (or multiply) an inequality by a negative number, we must reverse the direction of the inequality symbol. This is because dividing by a negative number changes the sign of both sides, and to maintain the correct relationship, we must flip the inequality sign.

Dividing both sides by -2 and reversing the inequality symbol, we get:

-4 / -2 ≥ -2x / -2

This simplifies to:

2 ≥ x

This inequality can also be written as:

x ≤ 2

This is the solution to the inequality. It means that 'x' can be any value less than or equal to 2. We have successfully isolated 'x' and determined the range of values that satisfy the original inequality.

Interpreting the Solution

The solution to the inequality 2 ≤ (2/3)(5-x) is x ≤ 2. This means that any value of x that is less than or equal to 2 will satisfy the original inequality. Understanding how to interpret this solution is just as important as finding it. The solution can be represented in several ways:

1. Number Line Representation

A number line is a visual way to represent the solution set. Draw a number line and mark the point 2. Since the solution includes 2 (x ≤ 2), we use a closed circle or a bracket at 2 to indicate that it is included in the solution set. Then, we shade the portion of the number line to the left of 2, representing all values less than 2. This shaded region visually represents the infinite number of solutions to the inequality. The number line representation provides an intuitive understanding of the range of possible values for x.

2. Interval Notation

Interval notation is another common way to express the solution set. In interval notation, we use parentheses and brackets to indicate whether the endpoints are included or excluded. For x ≤ 2, the solution set includes all numbers from negative infinity up to and including 2. We use a parenthesis for negative infinity because infinity is not a specific number and cannot be included. We use a bracket for 2 because it is included in the solution set. Therefore, the interval notation for x ≤ 2 is (-∞, 2]. Interval notation is a concise and precise way to represent solution sets, particularly for more complex inequalities.

3. Set-Builder Notation

Set-builder notation is a more formal way to describe the solution set using set theory. It defines the set of all x such that x is less than or equal to 2. The notation is written as:

{x | x ≤ 2}

This is read as "the set of all x such that x is less than or equal to 2." Set-builder notation is particularly useful for representing solutions to inequalities when dealing with more abstract mathematical concepts and sets.

Common Mistakes to Avoid

When solving inequalities, it's essential to be aware of common mistakes that can lead to incorrect solutions. Avoiding these pitfalls will ensure accuracy and a deeper understanding of the process.

1. Forgetting to Reverse the Inequality Sign

The most common mistake is forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number. As highlighted earlier, this is a crucial rule. Failing to reverse the sign will lead to an incorrect solution set. For example, if we have -2x > 4, dividing both sides by -2 should result in x < -2, not x > -2. Double-checking this step is vital for ensuring the correctness of your solution.

2. Incorrectly Distributing Constants

Another common mistake is incorrectly distributing constants, especially when dealing with parentheses. Ensure that the constant is multiplied by every term inside the parentheses. For example, if we have 3(x - 2) < 9, we must distribute the 3 to both x and -2, resulting in 3x - 6 < 9. A mistake in distribution can alter the entire inequality and lead to a wrong solution.

3. Misinterpreting the Inequality Symbols

Misinterpreting the inequality symbols can also lead to errors. Remember that '<' means “less than,” '>' means “greater than,” '≤' means “less than or equal to,” and '≥' means “greater than or equal to.” Confusing these symbols can lead to incorrect representations of the solution set. For example, understanding the difference between x < 3 and x ≤ 3 is crucial for accurately graphing the solution on a number line or expressing it in interval notation.

4. Not Checking the Solution

Finally, a good practice is to check your solution by plugging in values from the solution set back into the original inequality. This helps verify that your solution is correct. Choose a value within your solution set and a value outside of it to confirm that the inequality holds true for the former and not the latter. This step can catch any errors made during the solving process.

Real-World Applications of Inequalities

Inequalities are not just abstract mathematical concepts; they have numerous real-world applications across various fields. Understanding how to solve and interpret inequalities can provide valuable insights into practical problems.

1. Budgeting and Finance

In personal finance, inequalities are used to manage budgets and expenses. For example, if you have a monthly budget of $2000, you can express this as an inequality: Expenses ≤ $2000. This helps you track your spending and ensure that it stays within your financial limits. Inequalities can also be used to determine how much you need to save each month to reach a financial goal or to calculate the range of possible investment returns based on different risk levels.

2. Physics and Engineering

In physics, inequalities are used to describe constraints and limitations. For instance, the speed of an object cannot exceed the speed of light, which can be expressed as v < c, where v is the object's speed and c is the speed of light. Engineers use inequalities to design structures that can withstand certain loads or to ensure that systems operate within safe parameters. For example, the maximum stress a bridge can handle can be expressed as an inequality to ensure its structural integrity.

3. Computer Science

In computer science, inequalities are used in algorithm design and optimization. For example, the time complexity of an algorithm can be expressed as an inequality to determine its efficiency. Inequalities are also used in machine learning to define constraints for model parameters and to optimize model performance. Additionally, inequalities play a crucial role in cryptography, where they are used to ensure the security and integrity of data.

4. Economics

In economics, inequalities are used to model supply and demand, analyze market equilibrium, and optimize resource allocation. For example, the quantity demanded of a product must be less than or equal to the quantity supplied for a market to be in equilibrium. Inequalities are also used to analyze income distribution and poverty levels, providing insights into economic disparities and social welfare policies.

Conclusion

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. In this article, we have provided a step-by-step guide to solving the inequality 2 ≤ (2/3)(5-x), emphasizing the importance of each step and the underlying principles. We discussed the crucial rule of reversing the inequality sign when multiplying or dividing by a negative number and highlighted common mistakes to avoid. Furthermore, we explored various methods of interpreting the solution, including number line representation, interval notation, and set-builder notation. By understanding these concepts and practicing problem-solving, you can confidently tackle inequalities and apply them to real-world scenarios. Mastering inequalities is not just about solving mathematical problems; it's about developing a powerful tool for critical thinking and problem-solving in various aspects of life.