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Real numbers are the foundation of most mathematical computations. They encompass all the numbers on the number line. These include:

• Natural numbers (1, 2, 3, ...)
• Whole numbers (0, 1, 2, 3, ...)
• Integers (..., -2, -1, 0, 1, 2, ...)
• Rational numbers (fractions like \(\frac{1}{2}\), \(\frac{3}{4}\))
• Irrational numbers (numbers that cannot be expressed as fractions, like \(\sqrt{2}\) or \(\pi\))

The set of real numbers is crucial in solving equations, such as inequalities, because they allow for a complete representation of possible solutions on the number line. When looking for solutions to an inequality, you are typically searching for the real-number values that satisfy the condition.

Solving inequalities involves finding all the possible values of a variable that make the inequality true. Understanding how to manipulate inequalities is pivotal for checking conditions involving ranges, thresholds, or limits.
To solve an inequality:

• Start by simplifying the expression on both sides of the inequality.
• Add, subtract, multiply, or divide both sides of the inequality by the same number. Note, if you multiply or divide by a negative number, you must reverse the inequality sign.
• Combine solutions if working with multiple inequalities.

In the context of absolute value inequalities, setting up equivalent compound inequalities is essential, as seen in transitioning from \(|x^2 - 3| < 1\) to two simultaneous inequalities: \(x^2 - 3 < 1\) and \(x^2 - 3 > -1\).

Absolute value helps measure the size or distance from zero of a number on the number line, without considering its direction. Formally, the absolute value of a number \(x\) is represented by \(|x|\) and described as:

• \(x\), if \(x\) is greater than or equal to zero
• \(-x\), if \(x\) is less than zero

When dealing with inequalities, absolute values allow you to express a range of values within a certain distance from a central point. For example, \(|x^2 - 3| < 1\) indicates that \(x^2\) is less than 1 unit away from 3. The absolute value inequality can be split into two separate inequalities, representing both sides of the number's position in relation to the central value.

Mathematical expressions are combinations of numbers, variables, and operators (like plus or minus) arranged in a structured way to represent quantities or operations. Expressions can be simple, such as \(x + 2\), or complex, involving exponents and multiple terms, like \(x^2 - 3\).
An expression does not have an equality or inequality sign, unlike equations or inequalities. However, when expressions are part of an inequality, they help define the range of solutions. In solving \(|x^2 - 3| < 1\), we dealt with expressions like \(x^2 - 3\) to understand how close \(x^2\) is to the target value of 3.
Understanding and manipulating expressions is vital for setting up and solving inequalities accurately, ensuring proper interpretation of numerical and algebraic relationships.