A4 APPENDIX Review Figure 7 b a (c) a 7 b a b (b) a = b a b (a) a 6 b Figure 5 Real number line 3r 1 2 –1 2 –1 2 0 -1 - -2 -3 Scale 1 unit O 2 units 2 Figure 6 1 2 3 Zero Negative real numbers Positive real numbers –1 2 –1 2 0 O -1 - - –3 2 –3 2 -2 -3 Zero-Product Property If ab 0, = then a b 0, 0 = = , or both equal 0. In Words If a product equals 0, then one or both of the factors is 0. The Distributive Property can be used to remove parentheses: x x x 2 3 2 2 3 2 6 ( ) + = + ⋅ = + The Zero-Product Property will be used to solve equations (Section A.6). For example, if x2 0 = , then 2 0 = or x 0 = . Since 2 0 ≠ , it follows that x 0 = . The Real Number Line Real numbers can be represented by points on a line called the real number line . There is a one-to-one correspondence between real numbers and points on a line. That is, every real number corresponds to a point on the line, and each point on the line has a unique real number associated with it. Pick a point on a line somewhere in the center, and label it O . This point, called the origin , corresponds to the real number 0. See Figure 5. The point 1 unit to the right of O corresponds to the number 1. The distance between 0 and 1 determines the scale of the number line. For example, the point associated with the number 2 is twice as far from O as 1. Notice that an arrowhead on the right end of the line indicates the direction in which the numbers increase. Points to the left of the origin correspond to the real numbers 1, 2, − − and so on. Figure 5 also shows the points associated with the rational numbers 1 2 − and 1 2 and with the irrational numbers 2 and .π DEFINITION Coordinate; Real Number Line The real number associated with a point P is called the coordinate of P , and the line whose points have been assigned coordinates is called the real number line . Now Work problem 23 The real number line consists of three classes of real numbers, as shown in Figure 6. • The negative real numbers are the coordinates of points to the left of the origin O . • The real number zero is the coordinate of the origin O . • The positive real numbers are the coordinates of points to the right of the origin O . 2 Graph Inequalities An important property of the real number line follows from the fact that, given two numbers a and b , either a is to the left of b , or a is at the same location as b , or a is to the right of b . See Figure 7. If a is to the left of b , then “ a is less than b ,” which is written a b. < If a is to the right of b , then “ a is greater than b ,” which is written a b. > If a is at the same location as b , then a b. = If a is either less than or equal to b , then a b. ≤ Similarly, a b ≥ means that a is either greater than or equal to b . Collectively, the symbols , , , < > ≤ and ≥ are called inequality symbols . Note that a b < and b a > mean the same thing. It does not matter whether we write 2 3 < or 3 2. > Furthermore, if a b < or if b a, > then the difference b a − is positive. Do you see why? NOTE Zero is neither positive nor negative. j
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