24 CHAPTER R Review of Basic Concepts R.3 Real Number Operations and Properties ■ Order on the Number Line ■ Absolute Value ■ Operations on Real Numbers ■ Exponents ■ Order of Operations ■ Properties of Real Numbers Order on the Number Line If the real number a is to the left of the real number b on a number line, then a is less than b, written a 6b. If a is to the right of b, then a is greater than b, written a 7b. See Figure 16. Statements involving these symbols, as well as the symbols less than or equal to, …, and greater than or equal to, Ú, are inequalities. The inequality a 6b 6c says that b is betweena and c because a 6b and b 6c. The inequality symbol must point toward the lesser number. Distance is 5. Distance is 5. –5 5 0 Figure 17 The algebraic definition of absolute value follows. Absolute Value Let a represent a real number. 0 a0 = e a −a if a #0 if a *0 That is, the absolute value of a positive number or 0 equals that number, while the absolute value of a negative number equals its negative (or opposite). 1 !20 –11 7 2 3 7 9 –!5 p –5 0 1 2 3 4 5 –4 –3 –2 –1 -25 is to the left of -11 7 on the number line, so -25 6 -11 7 , and 2 20 is to the right of p, indicating that 220 7p. Figure 16 Absolute Value The undirected distance on a number line from a number to 0 is the absolute value of that number. The absolute value of the number a is written 0 a0 . For example, the distance on a number line from 5 to 0 is 5, as is the distance from -5 to 0. See Figure 17. Therefore, 0 50 = 5 and 0 -5 0 = 5. Inequality Symbols Symbol Meaning Example 3 is not equal to 3≠7 * is less than -4 6 -1 + is greater than 3 7 -2 " is less than or equal to 6 … 6 # is greater than or equal to -8 Ú -10 NOTE Because distance cannot be negative, the absolute value of a number is always positive or 0.
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