Problem 8 - 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. $$ x^{2}+2<4 $$

Step-by-Step Solution

Answer

The elements that satisfy the inequality \(x^2 + 2 < 4\) are \(-1, 0, \frac{1}{2}, 1\).

Step 1: Understanding the Problem

We are given the inequality \(x^2 + 2 < 4\). Our task is to determine which elements from the set \(S = \{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\) satisfy this inequality.

Step 2: Simplifying the Inequality

First, simplify the inequality \(x^2 + 2 < 4\) by subtracting 2 from both sides, resulting in \(x^2 < 2\).

Step 3: Checking Each Element from Set S

We need to check each element of set \(S\) to see if it satisfies \(x^2 < 2\). We'll test each element individually:

Step 4: Testing Element: (-2)

Calculate \((-2)^2 = 4\). Since 4 is not less than 2, \(-2\) does not satisfy the inequality.

Step 5: Testing Element: (-1)

Calculate \((-1)^2 = 1\). Since 1 < 2, \(-1\) satisfies the inequality.

Step 6: Testing Element: (0)

Calculate \(0^2 = 0\). Since 0 < 2, \(0\) satisfies the inequality.

Step 7: Testing Element: \(\frac{1}{2}\)

Calculate \(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\). Since \(\frac{1}{4} < 2\), \(\frac{1}{2}\) satisfies the inequality.

Step 8: Testing Element: (1)

Calculate \(1^2 = 1\). Since 1 < 2, \(1\) satisfies the inequality.

Step 9: Testing Element: \(\sqrt{2}\)

Calculate \((\sqrt{2})^2 = 2\). Since 2 is not less than 2, \(\sqrt{2}\) does not satisfy the inequality.

Step 10: Testing Element: (2)

Calculate \(2^2 = 4\). Since 4 is not less than 2, 2 does not satisfy the inequality.

Step 11: Testing Element: (4)

Calculate \(4^2 = 16\). Since 16 is not less than 2, 4 does not satisfy the inequality.

Step 12: Recording the Satisfying Elements

The elements that satisfy \(x^2 < 2\) are \(-1, 0, \frac{1}{2}, 1\). These elements are the solution to the problem.

Key Concepts

Algebra

Algebra is a branch of mathematics that helps us understand and solve problems involving unknown quantities. In this exercise, we focus on an algebraic inequality: \(x^2 + 2 < 4\). Inequalities are like equations, but they show a relationship of less than or greater than, instead of equality.

• To solve an inequality, we look for values of the variable (in this case \(x\)) that make the inequality true.
• We simplified the inequality by performing the same operation on both sides. Here, we subtracted 2 from both sides to isolate the \(x^2\) term, leading to \(x^2 < 2\).
• This tells us that we are looking for values of \(x\) whose squares are less than 2.

Understanding these basic algebraic operations is essential in solving inequalities.

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, called elements, and their relationships. In this problem, we analyze the set \(S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\).

• The set \(S\) contains numbers that we test against the inequality \(x^2 < 2\).
• Each element of the set needs to be individually checked to determine if it satisfies the given condition.
• When we explore elements in a set, we use concepts like union, intersection, and subset to analyze groupings.

Set theory allows us to systematically approach the problem by testing each element and recording which satisfies the inequality.

Numerical Inequality

A numerical inequality is a statement about the relative size or order of two values. Our goal here was to determine which numbers from a specific set satisfy \(x^2 < 2\).

• This task involves checking each number against the inequality to see if it holds true.
• The solution process includes squaring each element and deciding whether the result is less than 2.
• The numbers from the set that satisfy the inequality are the solution to our problem.

Understanding numerical inequalities is about identifying which numbers fit the criteria and why they fit those criteria.