Solving For X In The Inequality 9(2x + 1) < 9x – 18
In the realm of mathematics, inequalities play a crucial role in defining ranges and boundaries for solutions. Often, we encounter inequalities that require careful manipulation and problem-solving skills to arrive at the correct answer. This article delves into the inequality 9(2x + 1) < 9x – 18, providing a comprehensive, step-by-step guide to finding the value of x that satisfies the given condition. We will not only solve the inequality but also explore the underlying concepts and principles involved, ensuring a thorough understanding of the process. This is essential for anyone looking to build a solid foundation in algebra and inequality solving. Understanding inequalities is not just about finding a numerical answer; it's about grasping the relationship between variables and the range of values that satisfy a given condition. So, let's embark on this mathematical journey together, unraveling the intricacies of this inequality and discovering the value of x that lies within its solution set.
Understanding Inequalities: Before we dive into solving the specific inequality, it's essential to understand what inequalities are and how they differ from equations. Unlike equations, which have a single solution or a finite set of solutions, inequalities represent a range of values. The basic inequality symbols are: < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). When solving inequalities, we aim to isolate the variable on one side, just like in equations. However, there's a crucial difference: when we multiply or divide both sides of an inequality by a negative number, we must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the number line. For example, if 2 < 4, then multiplying both sides by -1 gives -2 > -4. This fundamental rule is critical in ensuring that the solution set remains accurate throughout the solving process. Additionally, understanding the properties of inequalities, such as the transitive property (if a < b and b < c, then a < c), helps in simplifying complex inequalities and reaching the solution more efficiently. So, with these foundational concepts in mind, let's move on to dissecting our specific inequality and finding its solution.
The Importance of Step-by-Step Solutions: When tackling mathematical problems, especially inequalities, a step-by-step approach is paramount. Each step should be logically connected to the previous one, ensuring that the solution is not only correct but also understandable. This methodical approach not only minimizes errors but also enhances comprehension of the underlying mathematical principles. In the context of the inequality 9(2x + 1) < 9x – 18, each algebraic manipulation must be performed carefully, keeping track of the operations and their impact on the inequality sign. From distributing the constant to combining like terms and isolating the variable, every step contributes to the final solution. Furthermore, a step-by-step solution allows for easy identification of potential errors. If a mistake is made, it can be quickly located and corrected without having to redo the entire problem. This is particularly important in examinations or assessments where time is limited. By breaking down the problem into manageable steps, we not only arrive at the correct answer but also gain confidence in our problem-solving abilities. Therefore, the emphasis on a step-by-step approach is not just about getting the answer; it's about developing a robust problem-solving methodology that can be applied to a wide range of mathematical challenges. So, let's now apply this principle to our inequality and see how it unfolds.
Step-by-Step Solution of 9(2x + 1) < 9x – 18
1. Distribute the Constant
Our initial step involves distributing the constant 9 across the terms within the parenthesis on the left side of the inequality. This means multiplying 9 by both 2x and 1. This operation is based on the distributive property of multiplication over addition, which states that a(b + c) = ab + ac. Applying this property to our inequality, we get: 9 * 2x + 9 * 1 < 9x – 18. This simplifies to 18x + 9 < 9x – 18. Distributing the constant is a fundamental step in simplifying algebraic expressions and inequalities. It allows us to remove the parenthesis, making it easier to combine like terms and isolate the variable. In this case, distributing the 9 has transformed the inequality into a more manageable form, setting the stage for the next steps in the solution process. It's crucial to perform this step accurately, as any error in distribution will propagate through the rest of the solution, leading to an incorrect answer. Therefore, double-checking the distribution ensures that we are on the right track. With the constant distributed, we can now proceed to the next phase of the solution, which involves combining like terms and working towards isolating x.
2. Combine Like Terms
After distributing the constant, our next goal is to combine like terms to simplify the inequality further. Like terms are terms that have the same variable raised to the same power. In our inequality, 18x + 9 < 9x – 18, we have two terms with the variable x (18x and 9x) and two constant terms (9 and -18). To combine like terms, we need to move the terms with x to one side of the inequality and the constant terms to the other side. A common approach is to subtract 9x from both sides to get all the x terms on the left side: 18x + 9 - 9x < 9x – 18 - 9x. This simplifies to 9x + 9 < -18. Next, we subtract 9 from both sides to isolate the x term further: 9x + 9 - 9 < -18 - 9. This simplifies to 9x < -27. Combining like terms is a crucial step in solving any algebraic equation or inequality. It reduces the complexity of the expression, making it easier to isolate the variable. By carefully moving terms across the inequality while maintaining the balance, we gradually work towards the solution. This step not only simplifies the inequality but also brings us closer to the final answer. So, with the like terms combined, we are now in a position to isolate the variable and find the solution set.
3. Isolate the Variable
Now that we have simplified the inequality to 9x < -27, the final step is to isolate the variable x. To do this, we need to get x by itself on one side of the inequality. Since x is being multiplied by 9, we can isolate it by dividing both sides of the inequality by 9. It's crucial to remember the rule that when dividing (or multiplying) both sides of an inequality by a negative number, we must flip the inequality sign. However, in this case, we are dividing by a positive number (9), so we do not need to flip the sign. Dividing both sides by 9, we get: (9x) / 9 < (-27) / 9. This simplifies to x < -3. Therefore, the solution to the inequality is x is less than -3. This means that any value of x that is less than -3 will satisfy the original inequality. Isolating the variable is the culmination of the previous steps, where we systematically simplified the inequality by distributing constants and combining like terms. This final step reveals the solution set, which in this case is all values of x less than -3. This solution provides a clear understanding of the range of values that satisfy the given condition. With the variable isolated, we have successfully solved the inequality and can now interpret the solution in the context of the original problem.
Determining the Value of x from the Given Options
Now that we have found the solution to the inequality 9(2x + 1) < 9x – 18, which is x < -3, we need to determine which of the given options falls within this solution set. The options provided are –4, –3, –2, and –1. To find the correct value of x, we need to identify which of these numbers is less than -3. Let's analyze each option:
- –4: –4 is less than –3, so it satisfies the inequality x < -3.
- –3: –3 is not less than –3 (it is equal to –3), so it does not satisfy the inequality.
- –2: –2 is greater than –3, so it does not satisfy the inequality.
- –1: –1 is greater than –3, so it does not satisfy the inequality.
Therefore, the only value of x from the given options that is in the solution set of the inequality is –4. This process of checking each option against the solution set is essential to ensure that we select the correct answer. It reinforces the understanding of what the solution set represents – all values that make the inequality true. By systematically evaluating each option, we can confidently identify the value of x that satisfies the given condition. This step-by-step approach not only leads to the correct answer but also strengthens our ability to interpret and apply the solution of an inequality in a practical context.
Conclusion: The Solution and Its Significance
In conclusion, by meticulously following the steps of distribution, combining like terms, and isolating the variable, we have successfully solved the inequality 9(2x + 1) < 9x – 18. Our solution is x < -3, which signifies that any value of x less than -3 will satisfy the given inequality. When presented with the options –4, –3, –2, and –1, we determined that only –4 falls within the solution set. This exercise highlights the importance of a systematic approach to solving inequalities, as each step plays a crucial role in arriving at the correct answer. Understanding inequalities is a fundamental skill in mathematics, with applications spanning various fields, from algebra and calculus to economics and engineering. The ability to solve inequalities allows us to define ranges, set boundaries, and model real-world scenarios where constraints and limitations are present. Furthermore, the process of solving inequalities reinforces critical thinking and problem-solving skills. It teaches us to analyze conditions, manipulate expressions, and interpret solutions in a meaningful way. The significance of this skill extends beyond the classroom, empowering us to make informed decisions and solve problems in diverse contexts. Therefore, mastering the techniques for solving inequalities is not just about finding numerical answers; it's about developing a valuable toolset for navigating the complexities of the world around us.
By breaking down the problem into manageable steps and thoroughly explaining each step, we have not only found the solution but also gained a deeper understanding of the underlying mathematical principles. This approach is key to building confidence and competence in solving inequalities and other mathematical challenges. The solution x < -3 is not just a number; it represents a range of values that satisfy a specific condition, and the ability to find and interpret such solutions is a valuable asset in both academic and practical pursuits.
