Problem 7 - Algebra and Trigonometry

Question

\(1-8=\) Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ \frac{1}{x} \leq \frac{1}{2} $$

Step-by-Step Solution

Answer

The elements 2 and 4 satisfy the inequality.

Step 1: Identify the Inequality to Solve

The inequality we need to solve is \( \frac{1}{x} \leq \frac{1}{2} \). This inequality means that the reciprocal of \(x\) should be less than or equal to \(\frac{1}{2}\).

Step 2: Transform the Inequality

We can transform the inequality \( \frac{1}{x} \leq \frac{1}{2} \) into \( x \geq 2 \) by taking reciprocal of both sides, keeping in mind to change the inequality sign.

Step 3: Evaluate Each Element of S

We need to check which elements of \( S \) satisfy \( x \geq 2 \). The set \( S \) is: \( \left\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\right\} \).

Step 4: Test Each Value for the Inequality

- For \( x = -2, -1, 0 \): These cannot satisfy \( x \geq 2 \) as they are negative or zero.- For \( x = \frac{1}{2}, 1, \sqrt{2} \): \( \frac{1}{2} \approx 0.707, \sqrt{2} \approx 1.414 \) are less than 2.- For \( x = 2 \): Satisfies \( x \geq 2 \).- For \( x = 4 \): Clearly satisfies \( x \geq 2 \).Therefore, the values \( 2 \) and \( 4 \) satisfy the inequality.

Key Concepts

Reciprocal Inequality

Understanding reciprocal inequalities can be quite intuitive once you grasp the basics. In this context, you are dealing with an inequality that involves the reciprocal of a variable, such as \( \frac{1}{x} \leq \frac{1}{2} \). This means we need to consider the inverse or reciprocal of each value of \( x \) and see if it is smaller than or at most equal to \( \frac{1}{2} \). When taking the reciprocal of both sides of an inequality, it is crucial to remember that if both numbers are positive and we exchange their positions, the inequality sign will flip. This necessitates that inequalities change direction when reciprocals are taken, provided the values involved are non-zero and positive. This forms the basis for transforming inequalities such as \( x \geq 2 \) from \( \frac{1}{x} \leq \frac{1}{2} \). This transformation simplifies complex inequalities into simpler ones we can evaluate directly. Reciprocals have a fascinating property; they invert the size relationship – greater numbers become smaller under inversion, and vice versa.

Set Evaluation

Evaluating a set involves checking each element against the inequality we are solving. Here, we need to evaluate which elements from the set \( S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\} \) satisfy the inequality \( x \geq 2 \). It's important to go through each element carefully:

• For negative numbers and zero, such as \(-2, -1,\) and \(0\), they cannot satisfy \( x \geq 2 \) because these values are not even positive.
• For positive numbers less than 2, such as \(\frac{1}{2}, 1,\) and \(\sqrt{2}\), these don't meet our criteria because they are still below 2 when evaluated numerically.
• The values \(2\) and \(4\) can satisfy our inequality, as they are either equal to or greater than 2.

Set evaluation helps in delineating which numbers conform to a given condition, essential for solving inequalities effectively.

Inequality Transformation

Transforming inequalities is a powerful tool often used in mathematics to simplify problems. In the case of the inequality \( \frac{1}{x} \leq \frac{1}{2} \), transforming it to \( x \geq 2 \) not only makes it easier to evaluate but also makes it easier to understand which values will satisfy it. This transformation involves changes of position between terms and involves flipping the inequality sign, which provides clarity. This transformation process is valuable because:

• It reduces complex expressions into simpler, more manageable ones.
• The transformation is logically consistent across a range of problems, making it applicable in various contexts.

To ensure you're correctly transforming an inequality, remember to carefully account for sign changes and the behaviour of numbers under operations like reciprocals. Such operations, when correctly applied, make problem-solving more straightforward, helping you readily identify solutions in set evaluations.