Problem 5 - Algebra and Trigonometry

Question

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ 3-2 x \leq \frac{1}{2} $$

Step-by-Step Solution

Answer

The elements of \(S\) that satisfy the inequality are \(\sqrt{2}, 2,\) and \(4\).

Step 1: Identify the Inequality

The given inequality is \(3 - 2x \leq \frac{1}{2}\). We need to find the elements of the set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) that satisfy this inequality.

Step 2: Rearrange the Inequality

First, rearrange the inequality to solve for \(x\). Subtract 3 from both sides to get: \(-2x \leq \frac{1}{2} - 3\). Simplifying the right side gives us: \(-2x \leq -\frac{5}{2}\).

Step 3: Divide by Negative Coefficient

Since we need \(x\), divide both sides by -2. Remember to reverse the inequality symbol: \[x \geq \frac{5}{4}\]. This tells us that any element \(x\) in the set \(S\) that is greater than or equal to \(\frac{5}{4}\) will satisfy the inequality.

Step 4: Check Elements of Set

Now, we simply check each element of the set \(S\). The elements are: - \(-2, -1, 0, \frac{1}{2}, 1\) - these do not satisfy \(x \geq \frac{5}{4}\)- \(\sqrt{2} \approx 1.41\), \(2\), \(4\) - these satisfy \(x \geq \frac{5}{4}\)

Key Concepts

Set Theory and Elements

Set theory is a branch of mathematical logic that studies collections of objects, which we call sets. In this exercise, we are given the set \( S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\} \). A set is denoted by curly braces and consists of distinct elements. These elements are the values that we will check against the inequality. Understanding the concept of sets is crucial because it allows us to systematically examine each element by applying the given conditions or operations. When analyzing a set like \( S \), we can think about how each element interacts with a mathematical statement, such as an inequality.
To determine whether an element satisfies a condition, we need to test each element individually. Here, our task is to see which elements satisfy the specific inequality. By testing each element, we can create a new set containing the elements that meet the given criteria. This approach is foundational in mathematics, especially when dealing with solutions that involve constraints.

Solving Algebraic Inequalities

Inequality solving is important in algebra and involves finding all values of a variable that satisfy a statement. In our problem, the inequality is \(3 - 2x \leq \frac{1}{2}\). Solving inequalities often involves similar steps as solving equations, but it's crucial to remember that the direction of the inequality changes when multiplying or dividing by a negative number.
Here are the steps to solve our inequality:

• First, we rearrange the inequality to isolate the term with \(x\). We subtract 3 from both sides to get \(-2x \leq \frac{1}{2} - 3\).
• Next, we simplify the expression \(\frac{1}{2} - 3\), which leads us to \(-2x \leq -\frac{5}{2}\).
• Finally, we divide both sides by \(-2\). Because we are dividing by a negative number, we reverse the inequality sign, obtaining \(x \geq \frac{5}{4}\).

In this result, \(x\) needs to be greater than or equal to \(\frac{5}{4}\) to satisfy the inequality. This means we can focus on values in the set that meet this condition.

Using Mathematical Reasoning

Mathematical reasoning involves using logic to solve problems and validate conclusions. In this exercise, we use reasoning to determine which elements from a given set satisfy the inequality \(3 - 2x \leq \frac{1}{2}\). Once we have solved the inequality and determined that \(x \geq \frac{5}{4}\), we can apply reasoning to ensure our solution is correct.
After reaching the expression \(x \geq \frac{5}{4}\), we review each element of the set \(S\). Using logical reasoning, we distinguish which numbers meet the criterion. For example:

• Numbers like \(-2, -1, 0, \frac{1}{2}, 1\) clearly do not satisfy \(x \geq \frac{5}{4}\), because they are all less than \(\frac{5}{4}\).
• On the other hand, numbers like \(\sqrt{2} \approx 1.41, 2,\) and \(4\) do satisfy the inequality because they are greater than or equal to \(\frac{5}{4}\).

Employing mathematical reasoning helps verify that \(\sqrt{2}, 2,\) and \(4\) are indeed the elements of set \(S\) that make the original inequality true, ensuring that the solution is consistent and valid.