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The triangle inequality is a foundational principle in mathematics, particularly in the study of metric spaces like the set of real numbers. It states that for any two real numbers, say, a and b, the absolute value of their sum is less than or equal to the sum of their absolute values, formally written as
\[ |a + b| \: \: \leq |a| + |b| \].
This principle gets its name from the geometric fact that in any triangle, the length of one side must be less than the sum of the lengths of the other two sides.

Understanding with an example

Consider the numbers 3 and -2. According to the triangle inequality: \[ |3 + (-2)| \: \: \leq |3| + |-2| \] which simplifies to \[ |1| \: \: \leq 3 + 2 \]. As 1 is indeed less than or equal to 5, the inequality holds. The triangle inequality is essential for proving various mathematical concepts, including those relevant to absolute value properties and the relationship between distances in geometry.

The absolute value measures the distance of a number from zero on the real number line, irrespective of direction. It's denoted by two vertical bars surrounding the number or expression, such as \( |x| \). Some key properties of absolute values involve their interaction with other numbers:

• Non-negativity: \( |x| \: \: \geq 0 \) for any real number \( x \).
• Identity: \( |x| = x \) if \( x \: \: \geq 0 \) and \( |x| = -x \) if \( x < 0 \).
• Product: \( |ab| = |a||b| \), meaning the absolute value of a product is the product of the absolute values.
• Quotient: \( |\frac{a}{b}| = \frac{|a|}{|b|} \), as long as \( b \) is not zero.

These properties are vital when we manipulate equations and inequalities in algebra. They underpin proofs, like the one in the exercise which demonstrates the relationship between the absolute value of a reciprocal and the reciprocal of an absolute value.

Real numbers are the collection of all rational and irrational numbers, encompassing a wide array of value types including integers, fractions, decimals, and numbers like \( \pi \) and \( \sqrt{2} \). They can be plotted on a one-dimensional line, known as the real number line.

• Rational Numbers: Can be expressed as the quotient of two integers, like \( \frac{3}{4} \) or \( -\frac{5}{2} \).
• Irrational Numbers: Cannot be written as exact fractions, such as \( \pi \) or \( \sqrt{3} \), and have non-repeating, non-terminating decimal expansions.

Understanding real numbers is crucial for grasping concepts in algebra, such as solving equations and inequalities, and in calculus, such as finding limits and derivatives. The set of real numbers is fundamental to all of mathematical analysis.

Inequalities express the relationship between two values when they are not equal. An inequality indicates whether one number is greater than, less than, greater than or equal to, or less than or equal to another number, using symbols like \( > \), \( < \), \( \: \: \geq \), and \( \: \: \leq \).
When working with inequalities, it's important to remember:

• Adding or subtracting the same number from both sides of an inequality does not change the direction of the inequality. For example, if \( a > b \), then \( a + c > b + c \) for any real number \( c \).
• Multiplying or dividing both sides by a positive number preserves the inequality direction, while doing the same with a negative number reverses it. For instance, if \( a \) is less than \( b \), and \( c < 0 \), then \( ac \) is greater than \( bc \).

Understanding how inequalities work with the absolute value is particularly important, as it helps us handle absolute differences and relationships, as shown in parts (b) and (c) of the exercise.