Correct Representations Of The Inequality 6x ≥ 3 + 4(2x - 1)
In the realm of mathematics, inequalities play a crucial role in defining relationships between expressions. Inequalities help us understand the range of possible values that a variable can take while satisfying a given condition. In this comprehensive article, we will dissect the inequality 6x ≥ 3 + 4(2x - 1), exploring various transformations and representations to identify the correct options. Our goal is to provide a detailed, step-by-step analysis that will help you grasp the underlying principles and apply them to similar problems.
Understanding Inequalities
Before we dive into the specifics of the given inequality, let's establish a solid foundation by defining what inequalities are and how they differ from equations. Inequalities are mathematical statements that compare two expressions using symbols such as '≥' (greater than or equal to), '≤' (less than or equal to), '>' (greater than), and '<' (less than). Unlike equations, which assert the equality of two expressions, inequalities express a range of possible values.
When solving inequalities, we aim to isolate the variable on one side, much like solving equations. However, a critical distinction arises when multiplying or dividing both sides by a negative number. In such cases, the direction of the inequality sign must be reversed to maintain the truth of the statement. This is a fundamental rule that must be adhered to meticulously.
The ability to manipulate inequalities accurately is essential in various fields, including algebra, calculus, and optimization problems. A thorough understanding of these concepts empowers us to model real-world scenarios, make informed decisions, and solve complex problems.
Initial Inequality: 6x ≥ 3 + 4(2x - 1)
Let's begin by restating the inequality we aim to dissect: 6x ≥ 3 + 4(2x - 1). This inequality presents a linear expression on both sides, involving the variable 'x'. Our task is to simplify and transform this inequality to identify equivalent representations from the given options.
The first step in simplifying this inequality involves expanding the expression on the right side. This means distributing the '4' across the terms within the parentheses: (2x - 1). By doing so, we eliminate the parentheses and pave the way for further simplification.
Step 1: Expanding the Right Side
To expand the right side, we multiply each term inside the parentheses by '4':
4 * (2x) = 8x
4 * (-1) = -4
So, the expanded expression becomes 8x - 4. Now, we substitute this back into the original inequality:
6x ≥ 3 + 8x - 4
This step is crucial as it transforms the inequality into a more manageable form, allowing us to combine like terms and isolate the variable 'x'. Expanding the expression correctly ensures that we maintain the integrity of the inequality and avoid any errors in the subsequent steps.
Step 2: Combining Like Terms
Now that we have expanded the right side, we can simplify the inequality further by combining the constant terms on the right side: 3 and -4. Combining these terms gives us:
3 - 4 = -1
Substituting this back into the inequality, we get:
6x ≥ 8x - 1
This simplification step is vital because it reduces the number of terms in the inequality, making it easier to isolate the variable 'x'. By combining like terms, we streamline the expression and bring it closer to a form that allows us to solve for 'x'.
Step 3: Isolating the Variable
To isolate the variable 'x', we need to move all terms containing 'x' to one side of the inequality and constant terms to the other side. A common approach is to subtract 8x from both sides of the inequality:
6x - 8x ≥ 8x - 1 - 8x
This simplifies to:
-2x ≥ -1
Isolating the variable is a key step in solving inequalities. By moving all 'x' terms to one side, we set the stage for the final step, which involves solving for 'x' itself. This process requires careful attention to the rules of inequality manipulation, particularly when dealing with negative coefficients.
Step 4: Solving for x
The final step in solving for 'x' involves dividing both sides of the inequality by the coefficient of 'x', which is -2. However, we must remember the crucial rule: when dividing or multiplying an inequality by a negative number, we must reverse the direction of the inequality sign.
Dividing both sides by -2 and reversing the inequality sign, we get:
(-2x) / (-2) ≤ (-1) / (-2)
This simplifies to:
x ≤ 1/2
This result represents the solution set for the inequality. It indicates that 'x' can take any value less than or equal to 1/2. Understanding this solution is crucial for identifying the correct representations of the original inequality.
Alternative Representation: 1 ≥ 2x
Now, let's explore an alternative representation of the solution. We can multiply both sides of the inequality x ≤ 1/2 by 2:
2 * x ≤ 2 * (1/2)
This simplifies to:
2x ≤ 1
This inequality is equivalent to the solution we found earlier. To further align it with one of the given options, we can rewrite it as:
1 ≥ 2x
This representation is crucial as it directly corresponds to one of the options provided, demonstrating the equivalence of different forms of the same inequality.
Identifying Correct Representations
Having simplified the original inequality and derived its solution, we can now systematically evaluate the given options to identify the correct representations. This process involves comparing each option with our simplified form and ensuring that they are mathematically equivalent.
Option 1: 1 ≥ 2x
As we derived in the previous section, the inequality 1 ≥ 2x is a correct representation of the original inequality. This option is obtained by simplifying and rearranging the original inequality, confirming its validity.
Option 2: 6x ≥ 3 + 8x - 4
This option represents the inequality after expanding the parentheses on the right side but before combining like terms. As we showed in Step 1, this is a valid intermediate step in simplifying the inequality. Therefore, this option is also a correct representation.
Option 3: x ≤ 1/2
This inequality represents the solution set for 'x' that we derived in Step 4. It is the most simplified form of the inequality and accurately captures the range of values that 'x' can take. Therefore, this option is a correct representation as well.
Common Mistakes to Avoid
When working with inequalities, it's crucial to be aware of common mistakes that can lead to incorrect solutions. One of the most frequent errors is forgetting to reverse the inequality sign when multiplying or dividing by a negative number. This mistake can completely change the solution set and lead to wrong conclusions.
Another common mistake is incorrectly distributing terms when expanding expressions. It's essential to ensure that each term inside the parentheses is multiplied by the term outside. A failure to do so can result in an incorrect simplification and an inaccurate solution.
Finally, errors can occur when combining like terms. Care must be taken to ensure that only terms with the same variable and exponent are combined. Mixing up terms can lead to an incorrect simplification and an erroneous solution.
Real-World Applications of Inequalities
Inequalities are not just abstract mathematical concepts; they have numerous real-world applications. They are used extensively in optimization problems, where the goal is to find the best possible solution within certain constraints. For example, businesses use inequalities to maximize profits while minimizing costs.
Inequalities are also crucial in modeling real-world scenarios, such as determining the safe operating range for equipment or calculating the maximum load a structure can bear. They are used in various fields, including engineering, economics, and computer science, to make informed decisions and solve complex problems.
Conclusion
In this article, we have thoroughly dissected the inequality 6x ≥ 3 + 4(2x - 1), demonstrating the step-by-step process of simplifying and solving it. We identified the correct representations of the inequality, emphasizing the importance of adhering to the rules of inequality manipulation. By understanding these principles and avoiding common mistakes, you can confidently tackle similar problems and apply these concepts in various real-world scenarios. Mastering inequalities is a valuable skill that enhances your mathematical prowess and problem-solving abilities.
