Solving The Inequality 0.35x - 4.8 < 5.2 - 0.9x A Step-by-Step Guide

In this article, we will delve into the process of solving the linear inequality 0.35x - 4.8 < 5.2 - 0.9x. Our primary goal is to identify the specific interval that represents the solution set for this inequality. This involves understanding the fundamental principles of inequality manipulation and applying them systematically to isolate the variable x. We will meticulously walk through each step, providing a clear and concise explanation to ensure a comprehensive understanding of the solution process. By the end of this discussion, you will be equipped with the knowledge and skills necessary to confidently solve similar inequalities and interpret their solution sets.

Before diving into the specifics of our problem, let's first establish a firm understanding of what linear inequalities are and how they differ from linear equations. 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). Unlike linear equations, which seek to find a specific value that makes the two sides equal, linear inequalities aim to identify a range of values that satisfy the given condition.

Understanding linear inequalities is crucial for solving a wide range of mathematical problems, particularly in areas such as optimization, calculus, and real-world applications. They are used to model scenarios where quantities are not necessarily equal but fall within a certain range. For instance, we might use inequalities to represent budget constraints, temperature ranges, or speed limits. The key difference between solving an equation and an inequality lies in the potential for the inequality sign to flip when multiplying or dividing both sides by a negative number. This is a critical rule to remember to ensure accurate solutions.

To effectively solve linear inequalities, it's essential to master the basic algebraic operations while being mindful of the specific rules that govern inequalities. This includes combining like terms, isolating the variable, and paying close attention to the direction of the inequality sign. Graphical representation of solutions on a number line can provide a visual understanding of the solution set, making it easier to interpret and apply in various contexts. The process often involves several steps, each requiring careful attention to detail to avoid errors. In the following sections, we will demonstrate how to apply these principles to solve the given inequality.

Now, let's tackle the inequality 0.35x - 4.8 < 5.2 - 0.9x step-by-step. Our objective is to isolate x on one side of the inequality to determine the solution set.

Step 1: Combine x terms. To begin, we want to gather all the x terms on one side of the inequality. We can achieve this by adding 0.9x to both sides of the inequality:

0. 35x - 4.8 + 0.9x < 5.2 - 0.9x + 0.9x

This simplifies to:

1. 25x - 4.8 < 5.2

Step 2: Isolate the x term. Next, we need to isolate the term containing x. We can do this by adding 4.8 to both sides of the inequality:

2. 25x - 4.8 + 4.8 < 5.2 + 4.8

This simplifies to:

3. 25x < 10

Step 3: Solve for x. Finally, to solve for x, we divide both sides of the inequality by 1.25. Since 1.25 is a positive number, we do not need to flip the inequality sign:

x < 10 / 1.25

x < 8

Therefore, the solution to the inequality 0.35x - 4.8 < 5.2 - 0.9x is x < 8. This means that any value of x less than 8 will satisfy the inequality.

The solution x < 8 represents a range of values, and we need to express this range as an interval. In interval notation, this solution set includes all real numbers less than 8, but not including 8 itself. This is represented as the interval (-∞, 8).

Interpreting the solution set in interval notation is a fundamental skill in algebra and is essential for accurately representing the range of values that satisfy an inequality. The use of parentheses indicates that the endpoint is not included in the solution set, while brackets would indicate inclusion. In our case, the parenthesis next to 8 indicates that 8 itself is not a solution, but all numbers infinitely close to 8 from the left are. This notation provides a concise and precise way to communicate the range of possible values for the variable.

Understanding the graphical representation of the solution set on a number line can further clarify the concept. We would represent x < 8 by shading the portion of the number line to the left of 8 and using an open circle at 8 to indicate that it is not included. This visual aid reinforces the idea that the solution consists of all numbers strictly less than 8. The interval notation and the graphical representation are two complementary ways of understanding the solution set of an inequality.

Based on our step-by-step solution, we have determined that the solution set for the inequality 0.35x - 4.8 < 5.2 - 0.9x is x < 8. This corresponds to the interval (-∞, 8).

To identify the correct interval, we carefully examined the inequality and applied the rules of algebraic manipulation. We combined like terms, isolated the variable x, and ultimately arrived at the solution x < 8. This result directly translates to the interval (-∞, 8), which encompasses all real numbers less than 8. The process of verifying the solution by substituting values from the interval back into the original inequality can provide additional confidence in the accuracy of the answer. For example, choosing a value such as x = 0, which is clearly less than 8, and plugging it into the original inequality, we can confirm that it satisfies the condition.

In the context of multiple-choice questions, understanding how to translate an inequality into interval notation is critical. The interval notation provides a concise and standardized way to express the solution set, making it easy to compare and select the correct answer. Moreover, being able to visualize the interval on a number line can serve as a helpful check, ensuring that the chosen interval accurately represents the solution.

When solving inequalities, it's easy to make mistakes if you're not careful. One common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. This is a crucial rule to remember, as it directly impacts the direction of the inequality and the final solution set. Another frequent error is incorrectly combining like terms or making arithmetic errors during the simplification process. These mistakes can lead to an incorrect solution, so it's essential to double-check each step.

Common mistakes in solving inequalities often stem from a misunderstanding of the fundamental rules or a lack of attention to detail. For instance, students may mistakenly apply the rules of solving equations to inequalities without considering the differences, such as the sign-flipping rule. Another pitfall is rushing through the steps without carefully considering the implications of each operation. It's important to take your time and approach the problem systematically, ensuring that each step is logically sound.

To avoid these common pitfalls, it's helpful to adopt a methodical approach. First, carefully read the inequality and identify the key components. Then, follow the steps outlined earlier for solving inequalities, paying close attention to detail and checking your work along the way. If you're unsure about a particular step, revisit the basic rules and principles before proceeding. Another useful strategy is to substitute a test value from the solution set back into the original inequality to verify that it satisfies the condition. This can help catch errors and provide confidence in your answer.

In conclusion, by following the steps of combining like terms, isolating the variable, and carefully considering the direction of the inequality, we have successfully determined that the solution set to the inequality 0.35x - 4.8 < 5.2 - 0.9x is x < 8, which corresponds to the interval (-∞, 8). Understanding the principles of solving inequalities and accurately representing the solution set in interval notation are essential skills in mathematics.