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In mathematics, a *number set* is a collection of numbers that have something in common. When we discuss number sets, we refer to categories that include integers, rational numbers, real numbers, and more.

Each number set serves a unique purpose and includes different types of numbers. Here are some common number sets:

• **Natural Numbers**: Numbers that begin from 1 and continue without end (1, 2, 3, ...).
• **Whole Numbers**: All natural numbers including zero (0, 1, 2, 3, ...).
• **Integers**: Whole numbers including negative numbers (-3, -2, -1, 0, 1, ...).
• **Rational Numbers**: Numbers that can be expressed as a fraction (like \( \frac{2}{3} \) or \( 0.75 \)).
• **Irrational Numbers**: Non-repeating, non-terminating decimals, such as \( \sqrt{2} \) or \( \pi \).
• **Real Numbers**: Any number that can be found on the number line. This includes both rational and irrational numbers.

In our exercise, the set \( S \) contains an assortment of integers (like -5 and 3), rational numbers (like \( \frac{2}{3} \)), and an irrational number (\( \sqrt{5} \)). Understanding number sets helps identify what type of numbers you're working with, which can affect how you solve equations and inequalities.

Polynomials are mathematical expressions that involve a sum of powers in one or more variables multiplied by coefficients. They are used extensively in algebra because they can model a wide variety of functions.

A polynomial can be expressed as a linear combination of terms of the form:\[ ax^n + bx^{n-1} + ... + zx^0 \]where \(a, b, ... , z\) are coefficients and \(n\) is a non-negative integer.

Different types of polynomials include:

• **Constant Polynomial**: Consists of a single term, like \(7\).
• **Linear Polynomial**: The highest power is 1, like \(x + 2\).
• **Quadratic Polynomial**: The highest power is 2, like \(x^2 + x + 1\).
• **Cubic Polynomial**: The highest power is 3, like \(x^3 - x + 1\).

When working with polynomials, particularly in inequalities, you're often interested in finding the specific values of the variable that satisfy the inequality. In our exercise, the polynomial in question is \(x^2 + 2\), which is a very simple quadratic polynomial. Our task was to determine for which number in our set \(S\) the inequality \(x^2 + 2 < 4\) holds true.

Square root operations involve finding a number which, when multiplied by itself, gives the original number. The square root is often represented by the radical symbol \(\sqrt{}\).

Here are a few key points about square roots:

• The square root of a number \(x\) is written as \(\sqrt{x}\).
• If \(x = y^2\), then \(y\) is a square root of \(x\).
• Every positive number has two square roots: one positive and one negative. For example, \(4\) has square roots \(+2\) and \(-2\).
• The square root operation cancels out squaring. For instance, \(\sqrt{x^2} = x\).

In the context of our problem, one of the elements of set \(S\) is \(\sqrt{5}\). Calculating the square involves reversing the square root operation: \((\sqrt{5})^2 = 5\). Understanding square root operations is crucial in solving inequalities that include expressions with squares, like in our exercise, where we evaluated \(x^2\) for each element of \(S\) to see if it was less than 2.