Inequalities are fundamental mathematical expressions that allow us to determine the range of values that satisfy a certain condition. The inequality we are tasked with analyzing, (0.35x – 4.8 < 5.2 – 0.9x), presents an opportunity to explore the methods of isolating a variable and establishing an interval solution. This problem not only serves as a practical exercise in algebra but also reinforces the importance of solving inequalities in various real-world applications, such as economics and engineering.

Analyzing the Inequality: Steps to Isolate the Variable

To begin with, we need to systematically manipulate the given inequality to isolate the variable (x). The expression starts as (0.35x – 4.8 < 5.2 – 0.9x). The first step is to eliminate the term involving (x) from one side and bring it to the other side. By adding (0.9x) to both sides, we convert the expression to (0.35x + 0.9x – 4.8 < 5.2). This step simplifies the equation and highlights the importance of balancing both sides for the inequality to remain valid.

Next, we combine like terms. The terms involving (x) can be added together, resulting in (1.25x – 4.8 < 5.2). Following this, we will aim to isolate (x) by getting rid of the constant term on the left side. To do this, we add (4.8) to both sides, which yields (1.25x < 10). This crucial step emphasizes the importance of systematic solving techniques in isolating variables effectively.

Finally, we divide both sides of the inequality by (1.25) to find (x). This results in (x < 8). It is essential in the context of inequalities to remember that dividing by a positive number does not change the inequality's direction. This last transformation not only yields a solution but also reinforces the methodical approach of working through the inequality step by step.

Establishing the Interval: Solutions for x in Context

Having isolated the variable, we can now interpret our solution in the context of an interval. The result, (x < 8), tells us that any value of (x) that is less than (8) satisfies the original inequality. This interval can be expressed in interval notation as ((-∞, 8)). Understanding this notation is crucial, as it conveys the range of acceptable solutions in a concise and standardized format, making it easier for further applications and analysis in various fields.

In practical terms, the implications of this solution depend heavily on the context in which the inequality is applied. For example, if (x) represents a quantity in a production model, the solution indicates that we should not produce more than (8) units to maintain a certain threshold related to cost or profit conditions outlined by our original inequality. Therefore, the established interval carries significant weight in decision-making processes, guiding stakeholders to stay within viable limits.

Moreover, it's worth noting that while we have defined a clear upper limit for (x), there is no lower limit imposed by our inequality, as the solution extends infinitely in the negative direction. This aspect can be critical in interpretation, especially in scenarios that involve risk management, budgeting, or resource allocation, where knowing both upper and lower bounds can influence strategic planning and operational efficiency.

In conclusion, determining the interval solution for the inequality (0.35x – 4.8 < 5.2 – 0.9x) involves a series of methodical steps aimed at isolating the variable (x) and interpreting its implications within a particular context. The final result, (x < 8), underscores the necessity of precision in mathematical reasoning and its applicability to real-world scenarios. Understanding how to manipulate inequalities not only enhances our mathematical toolkit but also equips us with the analytical skills required to tackle complex problems across various disciplines. Thus, mastering inequalities is not just an academic exercise; it is a vital competency for informed decision-making and effective problem-solving in everyday life.