Problem 7 - Precalculus: Mathematics for Calculus

Question

Let \(S=\left\\{-5,-1,0, \frac{2}{3}, \frac{5}{6}, 1, \sqrt{5}, 3,5\right\\}\) Determine which elements of \(S\) satisfy the inequality. $$-2+3 x \geq \frac{1}{3}$$

Step-by-Step Solution

Answer

The elements that satisfy the inequality are 1, \(\sqrt{5}\), 3, and 5.

Step 1: Understand the Inequality

The problem is asking us to find which elements from set \(S\) satisfy the inequality \(-2 + 3x \geq \frac{1}{3}\). This inequality can be rearranged to find the values of \(x\) which satisfy it.

Step 2: Rearrange the Inequality

Add 2 to both sides of the inequality to isolate terms involving \(x\) on one side. This gives us \(3x \geq \frac{1}{3} + 2\).

Step 3: Simplify the Right Side

Add \(2\) to \(\frac{1}{3}\) by converting \(2\) to a fraction with the same denominator: \(2 = \frac{6}{3}\). Therefore, the inequality becomes \(3x \geq \frac{1}{3} + \frac{6}{3} = \frac{7}{3}\).

Step 4: Solve for x

Divide both sides of the inequality by 3 to solve for \(x\): \(x \geq \frac{7}{9}\). This describes the condition that any element from set \(S\) must meet in order to satisfy the inequality.

Step 5: Verify Each Element of S

Check each element in the set \(S = \{-5, -1, 0, \frac{2}{3}, \frac{5}{6}, 1, \sqrt{5}, 3, 5\}\) to see if it satisfies the inequality \(x \geq \frac{7}{9}\). We find that the elements satisfying this condition are \(1, \sqrt{5}, 3,\) and \(5\), as each of these is greater than or equal to \(\frac{7}{9}\), which is approximately 0.777.

Key Concepts

Understanding Set Elements

When we talk about set elements, we are referring to individual items within a collection called a set. In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. Sets are often described as a collection of numbers or other elements.
A set is usually denoted by a capital letter, like set \( S \). Elements of a set, like \(-5, -1, 0, \frac{2}{3},\) and so on, are enclosed in curly braces \( \{ \} \) to differentiate them. Each element within the set is unique, ensuring no repetitions.

• The order of elements in a set does not matter.
• Sets can contain various types of items, but in the context of this problem, they include numbers.

Understanding the elements of a set is crucial when solving problems because you need to identify which elements satisfy certain conditions or properties. Here, we are focusing on which specific elements from our given set \( S \) meet the inequality condition.

Solving Inequalities

To solve an inequality means to find all the values of a variable that make the inequality true. Inequalities express a relationship where one quantity is greater than or equal to another. In our problem, we have the inequality \(-2 + 3x \geq \frac{1}{3}\), and we need to find values of \(x\) from the given set that satisfy this condition.
The process of solving an inequality generally involves a few steps:

• Isolate the variable: Begin by rearranging the inequality to get the variable by itself. Here, we add 2 to both sides, resulting in \(3x \geq \frac{7}{3}\).
• Simplify the expression: Ensure both sides are in a comparable form, such as fractions with common denominators.
• Perform operations: In this case, dividing both sides by 3 yields \(x \geq \frac{7}{9}\).

Once we solve the inequality, we can check if any set elements satisfy it by substituting them back into the variable \(x\) and verifying they make the inequality statement true.

Applying Mathematical Reasoning

Mathematical reasoning is the logical thinking that allows us to arrive at conclusions from given information. It involves evaluating statements, making connections, and systematically solving problems, such as inequalities.
In our exercise, once you solve for \(x \geq \frac{7}{9}\), you need mathematical reasoning to decide which elements in the set \( S \) fulfill this condition. This requires comparing each element from the set with \(\frac{7}{9}\), approximately 0.777.

• Compare elements one by one. For example, \(1\) is greater than \(0.777\), so it satisfies the inequality.
• Estimate square roots: Recognize that \(\sqrt{5}\) is approximately 2.236, which clearly satisfies the inequality.
• This same logic applies to \(3\) and \(5\), both greater than \(\frac{7}{9}\).

By applying reasoning systematically, you verify the solution and ensure the list of elements that satisfy the inequality is correct. It is this reasoning that confirms the solution as consistent and complete.