Problem 4 - 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. $$ 2 x-1 \geq x $$

Step-by-Step Solution

Answer

The elements are \(1, \sqrt{2}, 2, \text{and } 4\).

Step 1: Identify the Inequality

The problem provided is an inequality: \[ 2x - 1 \geq x \]This inequality needs to be solved to find which elements of the set \( S \) satisfy it.

Step 2: Isolate the Variable

To isolate \( x \), subtract \( x \) from both sides of the inequality:\[2x - 1 - x \geq x - x\]This simplifies to:\[x - 1 \geq 0\]

Step 3: Add Constants to Both Sides

Add \( 1 \) to both sides of the inequality to solve for \( x \):\[x - 1 + 1 \geq 0 + 1\]Simplifying gives:\[x \geq 1\]

Step 4: Determine Elements of Satisfying Set

Find all elements in the set \( S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\} \) that satisfy \( x \geq 1 \). The elements are 1, \( \sqrt{2} \approx 1.414\), 2, and 4.

Step 5: Conclusion

The elements of the set \( S \) that satisfy the inequality \( 2x - 1 \geq x \) are \( \{1, \sqrt{2}, 2, 4\} \).

Key Concepts

Solving Inequalities

When dealing with inequalities, the objective is to find all the values of the variable that make the inequality true. Unlike equations, inequalities have solutions in a range or even infinite values, depending on the inequality in question. Let's break down the process of solving them step-by-step:

• First, identify the inequality. In our exercise, it is: \[2x - 1 \geq x\]
• Next, manipulate the inequality like an equation to isolate the variable. You can add, subtract, multiply, or divide both sides by a number, just remember that multiplying or dividing by a negative number reverses the inequality sign.
• In our example, you subtract \(x\) from each side, simplifying to \[x - 1 \geq 0\].
• Finally, solve for the variable. Add 1 to each side to get \[x \geq 1\].

This gives us the range of values that satisfy the inequality. With these steps, solving inequalities becomes straightforward.

Set Elements

Understanding set elements is crucial for determining which specific values satisfy an inequality from a given set. Sets are collections of distinct objects, in this case, numbers. In our exercise, we were given the set \( S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\} \).To solve the exercise, we need to find which elements of the set satisfy the inequality \( x \geq 1 \). Here’s how to do it:

• Identify each element of the set. Check each to see if it satisfies the inequality.
• Compare each element with the inequality condition. Only select those which fulfill the condition.
• For our example, test each number: \(1\), \(\sqrt{2}\), \(2\), and \(4\) all meet the condition, as they are greater than or equal to 1.

This results in a subset of set \( S \) containing the elements \(\{1, \sqrt{2}, 2, 4\}\) that satisfy our inequality.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations or inequalities. It is the backbone of problem-solving in algebra. Here's a look at how it was applied in solving our inequality:In our exercise:

• Start by transforming the original inequality \(2x - 1 \geq x\) by subtracting \(x\) from both sides. This simplifies the expression and isolates the variable, resulting in \(x - 1 \geq 0\).
• Next, add \(1\) to both sides to solve for \(x\), yielding \(x \geq 1\).

These steps utilize basic algebraic operations: subtraction and addition. Each step in the manipulation is crucial for simplifying the inequality and determining a straight path to the solution. By mastering these operations, you can address more complex problems consistently and effectively.