Solving Inequality 1/x + 3/(2x) ≥ 5 And Clarifying Y=3 Question

In the realm of mathematics, inequalities play a pivotal role in describing relationships between quantities that are not necessarily equal. Solving inequalities involves determining the set of values that satisfy the given condition. This article delves into the process of finding the truth set for the inequality $\frac{1}{x} + \frac{3}{2x} \geq 5$, providing a step-by-step explanation and exploring the underlying mathematical concepts.

Understanding Inequalities and Truth Sets

Before we embark on solving the specific inequality, let's establish a firm understanding of the fundamental concepts. An inequality is a mathematical statement that compares two expressions using inequality symbols such as >, <, ≥, or ≤. The truth set, also known as the solution set, is the set of all values that make the inequality true. These values, when substituted into the inequality, will satisfy the given condition. Finding the truth set is a core skill in algebra and calculus, essential for solving a wide range of problems in mathematics and its applications.

In our case, the inequality we aim to solve is $\frac{1}{x} + \frac{3}{2x} \geq 5$. This inequality involves rational expressions, which adds a layer of complexity compared to simple linear inequalities. We need to carefully consider the domain of the variable x, ensuring that we avoid values that would lead to division by zero, which is undefined in mathematics. The process of finding the truth set involves algebraic manipulation, critical point identification, and interval testing. We will explore these steps in detail in the subsequent sections.

Mastering the art of solving inequalities requires a strong grasp of algebraic principles and a methodical approach. The ability to manipulate expressions, identify critical points, and test intervals is crucial for accurately determining the truth set. This skill is not only fundamental in mathematics but also has applications in various fields such as engineering, economics, and computer science, where inequalities are used to model and solve real-world problems. By understanding the concepts and techniques involved, we can confidently tackle a wide array of inequality problems.

Step-by-Step Solution to the Inequality

To find the truth set of the inequality $\frac{1}{x} + \frac{3}{2x} \geq 5$, we need to follow a systematic approach. Here's a detailed breakdown of the steps involved:

1. Combine the Fractions

The first step is to combine the fractions on the left-hand side of the inequality. To do this, we need to find a common denominator. In this case, the common denominator is 2x. Therefore, we rewrite the fractions as follows:

$\frac{1}{x} + \frac{3}{2x} = \frac{2}{2x} + \frac{3}{2x} = \frac{2 + 3}{2x} = \frac{5}{2x}$

Now our inequality becomes:

$\frac{5}{2x} \geq 5$

This simplification makes the inequality easier to work with. Combining fractions is a crucial step in solving inequalities involving rational expressions, as it allows us to consolidate the terms and proceed with further algebraic manipulations. By finding a common denominator and adding the numerators, we transform the inequality into a more manageable form, paving the way for the subsequent steps in the solution process. This step highlights the importance of understanding fraction operations in solving algebraic inequalities.

2. Manipulate the Inequality

Next, we want to get all the terms on one side of the inequality and zero on the other side. This is a standard technique in solving inequalities, as it allows us to identify critical points and test intervals. To achieve this, we subtract 5 from both sides of the inequality:

$\frac{5}{2x} - 5 \geq 0$

Now, we need to combine the terms on the left-hand side into a single fraction. To do this, we rewrite 5 as a fraction with a denominator of 2x:

$\frac{5}{2x} - \frac{10x}{2x} \geq 0$

Combining the fractions, we get:

$\frac{5 - 10x}{2x} \geq 0$

This step is crucial because it transforms the inequality into a form where we can easily identify the critical points. By manipulating the inequality and combining the terms, we set the stage for determining the intervals where the inequality holds true. This process involves careful algebraic manipulation to ensure that the inequality remains balanced and the solution set is accurately determined. The ability to manipulate inequalities is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems.

3. Identify Critical Points

Critical points are the values of x where the expression on the left-hand side of the inequality is either equal to zero or undefined. These points divide the number line into intervals, which we will test later to determine the solution set. To find the critical points, we set the numerator and the denominator equal to zero.

Setting the numerator equal to zero:

$5 - 10x = 0$

$10x = 5$

$x = \frac{1}{2}$

Setting the denominator equal to zero:

$2x = 0$

$x = 0$

Thus, our critical points are x = 0 and x = \frac{1}{2}. These critical points are pivotal in determining the solution set of the inequality. They represent the values where the expression changes its sign, transitioning from positive to negative or vice versa. By identifying these points, we can divide the number line into intervals and analyze the behavior of the expression within each interval. This step is essential for accurately solving inequalities and understanding the range of values that satisfy the given condition.

4. Create a Sign Chart

A sign chart is a visual tool that helps us determine the sign of the expression $\frac{5 - 10x}{2x}$ in different intervals. We use the critical points to divide the number line into intervals. Our critical points are 0 and \frac1}{2}, so we have three intervals to consider (-∞, 0), (0, \frac{1{2}), and (\frac{1}{2}, ∞).

We create a table with these intervals and test a value within each interval to determine the sign of the numerator (5 - 10x), the denominator (2x), and the entire expression (\frac{5 - 10x}{2x}).

Interval	Test Value	5 - 10x	2x	(5 - 10x) / (2x)
(-∞, 0)	x = -1	+	-	-
(0, 1/2)	x = 1/4	+	+	+
(1/2, ∞)	x = 1	-	+	-

This sign chart is a powerful tool for visualizing the behavior of the expression across different intervals. By analyzing the signs of the numerator and denominator, we can deduce the sign of the entire expression. This information is crucial for identifying the intervals where the inequality holds true. The sign chart provides a clear and organized way to track the changes in the expression's sign, making it easier to determine the solution set of the inequality. It is a fundamental technique in solving inequalities and is widely used in mathematics and related fields.

5. Determine the Solution Set

We are looking for the intervals where $\frac{5 - 10x}{2x} \geq 0$. From the sign chart, we see that the expression is positive in the interval (0, \frac{1}{2}). Additionally, the expression is equal to zero when x = \frac{1}{2}. However, we must exclude x = 0 because the original expression is undefined at x = 0.

Therefore, the solution set is:

$\left\{x: 0 < x \leq \frac{1}{2}\right\}$

This means that the inequality holds true for all values of x that are greater than 0 and less than or equal to \frac{1}{2}. The solution set represents the range of values that satisfy the given condition, and it is a crucial concept in mathematics and its applications. Determining the solution set accurately requires a thorough understanding of the inequality and the steps involved in solving it. By carefully analyzing the sign chart and considering the critical points, we can confidently identify the solution set and express it in a concise and meaningful way.

Analyzing the Answer Choices

Now, let's compare our solution set with the given answer choices:

(a) $\left\{x: 0 \leq x \leq \frac{1}{2}\right\}$ (b) $\left\{x: 0 < x < \frac{1}{2}\right\}$ (c) $\left\{x: 0 \leq x < \frac{1}{2}\right\}$ (d) $\left\{x: 0 < x \leq \frac{1}{2}\right\}$ (e) $\left{x:-\frac{1}{2} Correct answer is (d).

Why (d) is the Correct Answer

Our solution set is $\left\{x: 0 < x \leq \frac{1}{2}\right\}$, which matches answer choice (d). This answer choice correctly captures the range of values that satisfy the inequality. It includes all values of x greater than 0 and less than or equal to \frac{1}{2}, which aligns with our analysis of the sign chart and the critical points. The strict inequality (>) at 0 indicates that 0 is not included in the solution set, as the original expression is undefined at x = 0. The inclusion of \frac{1}{2} in the solution set is indicated by the (≤) symbol, as the inequality holds true when x = \frac{1}{2}.

The correct answer choice reflects a thorough understanding of the inequality and the solution process. It demonstrates the ability to accurately interpret the sign chart and translate it into a meaningful solution set. The careful consideration of the critical points and the behavior of the expression in different intervals is evident in the selection of answer choice (d). This highlights the importance of a methodical and analytical approach to solving inequalities.

Why Other Options are Incorrect

• (a) $\left\{x: 0 \leq x \leq \frac{1}{2}\right\}$: This option includes 0 in the solution set, which is incorrect because the original expression is undefined at x = 0.
• (b) $\left\{x: 0 < x < \frac{1}{2}\right\}$: This option excludes \frac{1}{2} from the solution set, which is incorrect because the inequality holds true when x = \frac{1}{2}.
• (c) $\left\{x: 0 \leq x < \frac{1}{2}\right\}$: This option includes 0 in the solution set and excludes \frac{1}{2}, both of which are incorrect.
• **(e) $\left{x:-\frac{1}{2}

Conclusion: Mastering Inequalities

Solving inequalities is a fundamental skill in mathematics with wide-ranging applications. By following a systematic approach, we can accurately determine the truth set for complex inequalities like $\frac{1}{x} + \frac{3}{2x} \geq 5$. This process involves algebraic manipulation, critical point identification, sign chart analysis, and careful consideration of the domain of the variable. The ability to solve inequalities is essential for success in algebra, calculus, and various other fields that rely on mathematical modeling and problem-solving.

Throughout this article, we have emphasized the importance of understanding the underlying concepts and techniques involved in solving inequalities. From combining fractions to creating sign charts, each step plays a crucial role in accurately determining the solution set. By mastering these skills, you can confidently tackle a wide array of inequality problems and apply them to real-world scenarios. The journey of learning mathematics is a continuous process, and the ability to solve inequalities is a valuable tool in your mathematical toolkit.

The second question presented in the original query, "Given that $y=3Discussion category :mathematics", appears to be incomplete and lacks the full context needed to provide a meaningful response. To address this, we need to rephrase the question to make it clear and understandable. The phrase "3Discussion category :mathematics" suggests that the question might be related to a discussion or problem set within the realm of mathematics where y is equal to 3. However, without additional information, it is impossible to determine the specific question being asked.

To clarify, we need to know what we are supposed to do with the information $y=3$. Are we asked to solve an equation, evaluate an expression, or perform some other mathematical operation? The missing part of the question is crucial for providing a relevant and accurate answer. The phrase "Discussion category :mathematics" seems to be a tag or label indicating the subject area but does not contribute to the mathematical content of the question itself.

In order to provide a helpful response, let's consider some possible interpretations of the intended question. For example, it could be:

1. If $y = 3$, evaluate the expression $2y + 5$.
2. If $y = 3$, solve the equation $y + x = 7$ for x.
3. If $y = 3$, find the value of $y^2 - 2y + 1$.

These are just a few examples, and the actual question could be something entirely different. The key takeaway is that the original question is incomplete and requires additional information to be properly addressed. Without a clear understanding of the question, any attempt to provide an answer would be speculative and potentially misleading.

In mathematical problem-solving, clarity and precision are paramount. A well-defined question is essential for obtaining a meaningful and accurate solution. The original question, in its current form, fails to meet this requirement. Therefore, it is crucial to rephrase or complete the question before attempting to solve it. This ensures that the effort spent on finding a solution is directed towards the intended problem, and the answer obtained is relevant and correct.

Moving forward, it is important to ensure that mathematical questions are stated clearly and completely, including all necessary information and context. This promotes effective communication and facilitates the accurate application of mathematical principles and techniques. A well-defined question serves as the foundation for a successful problem-solving process, leading to a clear and concise solution.