Solving -3(2x-5) < 5(2-x) Correct Representations And Step-by-Step Solution
In the realm of mathematics, inequalities play a pivotal role in defining relationships between quantities that are not necessarily equal. They provide a powerful tool for expressing constraints, setting boundaries, and solving a wide range of problems across various disciplines. Among the different types of inequalities, linear inequalities hold a special significance due to their simplicity and widespread applicability. This article delves into the intricacies of solving a specific linear inequality, , while also exploring the underlying concepts and techniques involved. We will dissect the problem step by step, providing a comprehensive guide for readers to grasp the fundamental principles of solving inequalities.
Understanding Linear Inequalities
Before we embark on solving the given inequality, it's crucial to establish a solid understanding of linear inequalities themselves. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). These symbols indicate that the two expressions are not necessarily equal, but rather one is either smaller or larger than the other. Linear inequalities, in particular, involve expressions where the variable appears only to the first power, meaning there are no exponents or other non-linear operations applied to the variable.
The solution to a linear inequality is the set of all values of the variable that make the inequality true. Unlike linear equations, which typically have a single solution, linear inequalities often have a range of solutions. This range can be represented graphically on a number line, where the solution set is depicted as an interval or a union of intervals. Understanding the concept of solution sets is essential for interpreting and applying the results obtained from solving inequalities.
Step-by-Step Solution of the Inequality
Now, let's turn our attention to the specific inequality at hand: . To solve this inequality, we will follow a series of algebraic steps that aim to isolate the variable x on one side of the inequality symbol. Each step is carefully designed to maintain the balance of the inequality while progressively simplifying the expression.
1. Distribute the Constants
The first step in solving the inequality is to eliminate the parentheses by distributing the constants on both sides. This involves multiplying the constant outside the parentheses by each term inside the parentheses. On the left side, we have -3 multiplied by (2x-5), which yields -6x + 15. On the right side, we have 5 multiplied by (2-x), which results in 10 - 5x. Thus, after the distribution, the inequality becomes:
-6x + 15 < 10 - 5x
This step is crucial as it removes the grouping symbols and allows us to combine like terms in subsequent steps.
2. Combine Like Terms
The next step involves combining like terms, which are terms that contain the same variable raised to the same power. In this case, we have x terms and constant terms on both sides of the inequality. To combine the x terms, we can add 5x to both sides of the inequality. This will eliminate the -5x term on the right side and bring all the x terms to the left side. Adding 5x to both sides, we get:
-6x + 5x + 15 < 10 - 5x + 5x
Simplifying this expression, we obtain:
-x + 15 < 10
Now, to combine the constant terms, we can subtract 15 from both sides of the inequality. This will isolate the x term on the left side and move all the constant terms to the right side. Subtracting 15 from both sides, we get:
-x + 15 - 15 < 10 - 15
Simplifying this expression, we obtain:
-x < -5
3. Isolate the Variable
The final step in solving the inequality is to isolate the variable x. Currently, we have -x on the left side, which means we need to multiply or divide both sides of the inequality by -1. However, it's crucial to remember that multiplying or dividing an inequality by a negative number reverses the direction of the inequality symbol. This is a fundamental rule in solving inequalities and must be applied carefully.
Multiplying both sides of the inequality by -1, we get:
(-1)(-x) > (-1)(-5)
Notice that the less than symbol (<) has been reversed to a greater than symbol (>) due to the multiplication by -1. Simplifying this expression, we obtain:
x > 5
This is the solution to the inequality. It states that x is greater than 5, meaning any value of x that is larger than 5 will satisfy the original inequality.
Correct Representations of the Inequality
Now that we have solved the inequality, let's examine the given options and identify the correct representations. The original question asks us to select two options that accurately represent the inequality . Based on our step-by-step solution, we can now evaluate each option.
A. x < 5
This option is incorrect. Our solution clearly states that x > 5, meaning x is greater than 5, not less than 5.
B. -6x - 5 < 10 - x
This option is incorrect. It seems to be an attempt to distribute the constants, but there's an error in the distribution. The correct distribution of -3(2x-5) should be -6x + 15, not -6x - 5.
C. -6x + 15 < 10 - 5x
This option is correct. It represents the inequality after the constants have been correctly distributed on both sides, as we showed in step 1 of our solution.
Based on our analysis, the two correct representations of the inequality are:
- C. -6x + 15 < 10 - 5x
- Solution: x > 5 (derived from the correct simplification)
Graphical Representation of the Solution
To further solidify our understanding of the solution, let's consider the graphical representation of x > 5 on a number line. A number line is a visual tool that represents all real numbers as points on a line. To represent the solution x > 5, we draw a number line and mark the point 5. Since the inequality is strictly greater than (x > 5), we use an open circle at 5 to indicate that 5 itself is not included in the solution set. Then, we shade the portion of the number line to the right of 5, representing all numbers greater than 5. This shaded region represents the solution set of the inequality.
The graphical representation provides a clear visual depiction of the solution and helps to reinforce the concept that inequalities often have a range of solutions rather than a single value.
Importance of Solving Inequalities
Solving inequalities is a fundamental skill in mathematics with far-reaching applications in various fields. Inequalities are used to model real-world situations involving constraints, limitations, and optimization problems. For example, in economics, inequalities can be used to represent budget constraints or resource limitations. In engineering, they can be used to define safety margins or tolerance levels. In computer science, inequalities are used in algorithms and optimization problems.
Understanding how to solve inequalities is crucial for making informed decisions and solving problems in these diverse fields. The ability to manipulate inequalities, isolate variables, and interpret solution sets is a valuable asset for anyone pursuing studies or careers in STEM fields.
Common Mistakes to Avoid
While solving inequalities may seem straightforward, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and ensure accurate solutions.
- Forgetting to Reverse the Inequality Symbol: The most common mistake is forgetting to reverse the inequality symbol when multiplying or dividing both sides by a negative number. This rule is crucial for maintaining the balance of the inequality and obtaining the correct solution.
- Incorrect Distribution: Another common mistake is making errors during the distribution step. Ensure that you multiply the constant outside the parentheses by each term inside the parentheses, paying careful attention to the signs.
- Combining Unlike Terms: Avoid combining terms that are not like terms. Only terms with the same variable raised to the same power can be combined.
- Misinterpreting the Solution Set: Pay close attention to the inequality symbol and correctly interpret the solution set. For example, x > 5 means x is greater than 5, while x ≤ 5 means x is less than or equal to 5.
By being mindful of these common mistakes, you can significantly improve your accuracy in solving inequalities.
Conclusion
In this comprehensive guide, we have explored the process of solving the linear inequality step by step. We began by understanding the concept of linear inequalities and their solution sets. Then, we meticulously worked through the algebraic steps, including distributing constants, combining like terms, and isolating the variable. We emphasized the importance of reversing the inequality symbol when multiplying or dividing by a negative number.
Furthermore, we identified the correct representations of the inequality from a set of options and discussed the graphical representation of the solution on a number line. We also highlighted the significance of solving inequalities in various fields and common mistakes to avoid.
By mastering the techniques presented in this guide, readers will gain a solid foundation in solving linear inequalities and be well-equipped to tackle more complex mathematical problems. The ability to solve inequalities is a valuable skill that extends beyond the classroom and into numerous real-world applications.
By following the steps and understanding the concepts outlined in this article, you can confidently solve linear inequalities and apply this knowledge to a wide range of mathematical and real-world problems. Remember to practice regularly and pay attention to the details, and you will master the art of solving inequalities.
