Determining the Solution Set for 0.35x – 4.8 < 5.2 – 0.9x

Inequalities are an essential part of algebra that allow us to express relationships between quantities that are not necessarily equal. In this article, we will explore the inequality (0.35x – 4.8 < 5.2 – 0.9x). Understanding how to manipulate and solve such inequalities is useful, not just in mathematics, but also in fields such as economics, engineering, and the sciences, where constraints or limits are often modeled using inequalities. By carefully analyzing the steps involved in isolating the variable (x), we can determine the solution set and its implications.

Analyzing the Inequality: Steps to Isolate x Effectively

To solve the inequality (0.35x – 4.8 < 5.2 – 0.9x), we begin by rearranging the terms to isolate (x) on one side. First, we can add (0.9x) to both sides to eliminate the (x) term from the right side of the inequality. This yields:

[
0.35x + 0.9x – 4.8 < 5.2
]

Combining the (x) terms on the left results in:

[
1.25x – 4.8 < 5.2
]

Next, we add (4.8) to both sides to further isolate the term containing (x):

[
1.25x < 5.2 + 4.8
]

This simplifies to:

[
1.25x < 10
]

Now, to find (x), we divide both sides by (1.25):

[
x < frac{10}{1.25}
]

Calculating the right-hand side gives:

[
x < 8
]

Through these systematic steps, we have effectively isolated the variable (x) and identified the critical point at which the inequality holds true.

Evaluating the Solution Set: Implications and Interpretations

The solution set (x < 8) indicates that any value of (x) less than (8) satisfies the original inequality (0.35x – 4.8 < 5.2 – 0.9x). This outcome has significant implications, particularly when considered in various contexts. For instance, if (x) represents a quantity such as time, resources, or financial investments, the restriction implies a limit on the maximum feasible value. Understanding this upper bound allows for better decision-making based on the available data.

Additionally, one must consider the nature of inequalities, as they often represent ranges or intervals of possible outcomes rather than fixed points. This inherent flexibility can be advantageous in practical applications, such as optimizing production levels or managing constraints in budget allocations. By interpreting the solution set as an open interval extending from negative infinity up to, but not including, (8), stakeholders can explore a wide array of options that comply with the established limit.

Lastly, this analysis emphasizes the importance of inequalities in formulating policies or strategies. Recognizing the constraints imposed by (x < 8) can lead to more strategic planning and resource allocation. Decision-makers can utilize these insights to devise plans that remain within the bounds of the inequality, ensuring that their actions are both feasible and effective.

In conclusion, solving the inequality (0.35x – 4.8 < 5.2 – 0.9x) involves a systematic approach to isolating the variable and understanding the significance of the resulting solution set. The conclusion (x < 8) provides critical insights into limits and constraints that can be applied across various fields. The ability to interpret these inequalities enables more informed decisions, ensuring that actions taken align with the established parameters. Overall, mastering the techniques of solving inequalities is essential for both academic and real-world applications.