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Magazine Chapter 1: Problem 8
\(1-8=\) Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ x^{2}+2<4 $$
Short Answer
Expert verified
The elements that satisfy the inequality \(x^2 + 2 < 4\) are \(-1, 0, \frac{1}{2}, 1\).
Step by step solution
01
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.
02
Simplifying the Inequality
First, simplify the inequality \(x^2 + 2 < 4\) by subtracting 2 from both sides, resulting in \(x^2 < 2\).
03
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:
04
Testing Element: (-2)
Calculate \((-2)^2 = 4\). Since 4 is not less than 2, \(-2\) does not satisfy the inequality.
05
Testing Element: (-1)
Calculate \((-1)^2 = 1\). Since 1 < 2, \(-1\) satisfies the inequality.
06
Testing Element: (0)
Calculate \(0^2 = 0\). Since 0 < 2, \(0\) satisfies the inequality.
07
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.
08
Testing Element: (1)
Calculate \(1^2 = 1\). Since 1 < 2, \(1\) satisfies the inequality.
09
Testing Element: \(\sqrt{2}\)
Calculate \((\sqrt{2})^2 = 2\). Since 2 is not less than 2, \(\sqrt{2}\) does not satisfy the inequality.
10
Testing Element: (2)
Calculate \(2^2 = 4\). Since 4 is not less than 2, 2 does not satisfy the inequality.
11
Testing Element: (4)
Calculate \(4^2 = 16\). Since 16 is not less than 2, 4 does not satisfy the inequality.
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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.
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.
