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The Number Line, Absolute Value, Inequalities, and Properties of R I. Homework II. The Number Line To draw a number line we draw a line with several dashes in it and ordered numbers below the line, both positive and negative. The number corresponding to the point on the number line is called the coordinate of the number line. This will be demonstrated in class. III. Absolute Values Absolute value signs make the inside positive: |-3| = 3, |2| = 2, |4 - 1| = |3| = 3 We can write |x| =
IV. Inequalities Recall the four inequalities:
When we graph an inequality on a number line we use "[" or "]" to include the point and "(" or ")" to not include the point. Example: Graph the following on a number line A) {x|x < 3} B) {x|x > 2} C) {x|3 < x < -5} D) {x||x| > 4} VI) Properties of Addition and Multiplication A) Commutative Property Addition: a + b = b + a Multiplication: ab = ba B) Associative Property Addition: (a + b) + c = a + (b + c) Multiplication: (ab)c = a(bc) C) Identity Property Addition: there is a 0 such that a + 0 = 0 + a = a Multiplication: there is a 1 such that a1 = 1a = a D) Inverse Property Addition: for any a, there is a -a with a + -a = 0 Multiplication: for any a not 0, there is a 1/a with a(1/a) = 1 E) Distributive Property a(b + c) = ab + ac (a + b)c = ac + bc F) Trichotomy If a and b are real numbers then one of the three must hold i) a < b ii) a > b iii) a = b G) Transitivity If a, b, and c are real numbers and a < b and b < c then a < c.
We will play the "list the property game": (2 + 3) + 4 = 2 + (3 + 4) (Associative Property of Addition) (x - y)(x + y) = (x + y)(x - y) (Commutative Property of Multiplication) (2 - y)[1/(2 - y)] = 1 (Multiplicative inverse) Other examples will be given. Exercises: complete the following A) x - z = ________ (Commutative) B) w(0) = _________ (Multiplication property of 0) C) 3(xy) = __________ (Associative) D) x(y - 3) = _________(Distributive)
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