Absolute Value and Opposites in Mathematics

Absolute Value of a Number

• The absolute value of a number is the distance between that number and zero on the number line.
• The absolute value of a number is never negative.
• It is denoted by two vertical bars (| |).
• For example, |2| = 2, |-2| = 2, |0| = 0

Opposites (Additive Inverses)

• Opposites are numbers that are equidistant from zero on the number line but have opposite signs.
• For example, the opposite of 3 is -3, the opposite of -5 is 5.
• The opposite of a number is also called its additive inverse.

Examples of Opposites

• Opposites of numbers: 3 and -3; -10 and 10; 56 and -56; 49 and -49
• Opposites of words/concepts: cold and warm; down and up; north and south; rising and falling; decrease and increase; backward and forward; below and above

Number Line

• A number line visually represents numbers in order from smallest to largest.
• Positive numbers lie to the right of zero on the number line; negative numbers lie to the left.
• Opposites are found on opposite sides of zero, equally distant from it.

Absolute Value Examples

• |4| = 4
• |-4| = 4
• |2+3|-|4-3| =5-1 = 4
• |7-4|-|5-3| =3-2 = 1 (Solved example of absolute value operation)

Comparing Absolute Values

• Symbols like < (less than), > (greater than), and = (equal to) are used to compare the absolute values of different numbers.
• For example: |2| > |-2|; |10 - 2| = |6 + 2|; |-5-2| < |-1 + 1| ; |8 -3| = |6 - 1|