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# 0.3 notes - Positive and negative numbers I A number is...

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Unformatted text preview: Positive and negative numbers: I A number is called positive if it lies to the right of 0 on the real line. In A number is called negative if it lies to the left of 0 on the real line. Examine algebraic properties on p. 18 . We say that a number a is less than a number b, written a s: b, if :1 lies to the left ofb on the real line. We say that a is less than or equal to b, a Eb, ifa <1 b or a = b. Similarly, we say that a is greater than 2?, written a 3* b, if a lies to the right of b on the real line. Thus a > b means the same as b < a. Transitivity: lfa<bandb<c, thenasic. Often the multiple inequalities are written together as a single string of inequalities. Thus a < b < c means the same thing as a <1 b and b <1 c. Addition of inequalities: lfa {bandcéd,theno+c<:b +d. Multiplication of an inequality: Suppose a <1 b. lfc 3’ 0, then ac < be. Ifc < 0, then ac 3* be. Sets: A set is a collection of objects (elements). If a set contains only finitely many objects, then the objects in the set can be explicitly displayed between the symbols { }. For example, {8, 29, 2008}. (This is in the roster form.) Sets can also be denoted by a property that characterizes objects of the set. For example, {I : x > 2}. (This is in the set-builder form.) Here the notation should read "the set of real numbers x such that x is greater than 2." Interval: An interval is a set of real numbers that contains all numbers between any two numbers in the set. Examine interval notations on p. 22. Intervals (continued): Sometimes we need to use intervals that extend arbitrarily far to the leﬁ or to the right on the real number line. Examine interval notations on p. 22. Here the symbol no , called iuﬁuirv, should be thought of simply as a notational convenience. Union: The union of two sets A and B, denoted A U3, is the set of objects that are contained in at least one of the sets A and B. Write (l, 5) U (3, 7'] as an interval. Write the set of nonzero real numbers as the union of two intervals. Absolute value: The absolute vut’ue of a number is its distance from 0. The absolute value of a number I), denoted lb|, is deﬁned by bfbao lb|= —a new. Care should be taken to apply this rule only to numbers, not to expressions whose value is unknown. Write the inequality |x| < 2 without using an absolute value. Write the set [xrlxl «t 2] as an interval. Write the inequality lx— Sl <1 without using an absolute value. Write the set [x: b:— 5| <1] as an interval. Suppose b is a real number and h I? 0. (3) Write the inequality |x— b| < 1% without using an absolute value. (b) Write the set [x: |x—b| dz] as an interval. ...
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0.3 notes - Positive and negative numbers I A number is...

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