Math 1351 011 Syllabus Distance on a number line ordering Absolute value Distance in the plane midpoint formula Homework 1 Distance on a number line Real numbers Real number line Distance on a number line 2 Distance from x to the origin Distance between two points on the line 1 0 1 Order properties real numbers a b c d Trichotomy law exactly one of the following is true a b a b or a b Transitive law of inequality if a b and b c then a c Additive law of inequality if a c and b d then a b c d Multiplicative law of inequality if a b then ac bc if c 0 and ac bc if c 0 2 Absolute value a The number x is located x units from 0 To the right if x 0 to the left if x 0 Distance between numbers x1 and x2 is x1 x2 a if a 0 a if a 0 Note that x1 x2 x2 x1 Terminology p if and only if q means that both the statement and its converse are true i e if p then q and if q then p Properties of the absolute value For a b real numbers a 0 absolute value is nonnegative a a a 2 a2 ab a b a b a b b 0 a a a Let b 0 a b if and only if a b Let b 0 a b if and only if b a b Let b 0 a b if and only if a b or a b a b a b true because either a a or a a think of number line Triangle Inequality very useful in theory and computation Interval notation Closed interval Open interval Half open Real number line a x b a b a x a x b b a x b a b a x a x b b a x b a b a x b a b Absolute value equations Example solve 2x 6 x Solution if 2x 6 0 then 2x 6 2x 6 solve 2x 6 x x 6 if 2x 6 0 then 2x 6 2x 6 solve 2x 6 x 3x 6 x 2 Two solutions x 6 and x 2 It is always a good idea to plug them back in and doubled check Recall that a b is the distance between a and b on the number line So x a b is satisfied by the two points x that are of distance b from a The equation above is satisfied by the two points x such that the distance between 2x and 6 on the number line is x Absolute value inequality Example Solve 2x 3 4 Solution 4 2x 3 4 using one of our abs value properties 4 3 2x 4 3 1 2x 7 1 2 x 7 2 1 2 7 2 in interval notation See textbook for geometric solution Absolute value as tolerance Let w be a measurement e g weight w a b can be interpreted as w being compared to a with absolute error of measurement of b units Example a bag of cement weighs 90 lbs plus or minus 2 lbs So a given bag can weigh as much as 92 lbs or as little as 88 lbs State as an absolute value inequality Solution let w weight of the bag of cement in pounds 90 2 w 90 2 2 w 90 2 w 90 2 Distance between points in the plane Theorem distance between two points P1 x1 y1 and P2 x2 y2 is given 2 2 2 2 by d x y x2 x1 y 2 y1 x horizontal change x2 x1 a k a run y vertical change y2 y1 a k a rise Proof d2 x 2 y 2 Pythagorean theorem d2 x 2 y 2 abs value property d x y 2 P2 d 2 y2 y1 y P1 x2 x1 x Midpoint formula Midpoint of a line segment with endpoints P1 x1 y1 and P2 x2 y2 has coordinates M x1 x2 x1 x2 2 2 st nd Average 1 and 2 components of coordinates of endpoints
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