algebra1 - N UMBER S YSTEMS T he n atural numbers are t he...

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z og u « ~ E o. <1> '" ill "0 ,..; 15 g ... ~ ~ C- ] '" o. ~ .~ .0 0 0> ~ ... N-o~ut 8~O:::l g N ~.!!! g) :c @(J)~Oaa u ~;~~ ~ "0 c 'C"'E,~ 0 <1> ~.>:: (; SttS c 84 Sto c( ~ -~x - 9 + 4x = 36 - 4 (x - ~) . Combine like terms on the left side: ~x - 9 = 36 - 4 (x - ~) NUMBER SYSTEMS The natural numbers are the numbers we count with: 1, 2. 3, 4, 5, 6, , 27, 28, .. The whole numbers are the numbers we count with and zero: 0, 1, 2, 3, 4, 5, 6, . The integers are the numbers we count with, their negatives. and zero: ... , -3, -2, -1, 0, I, 2, 3, . -The positive integers are the natural numbers. -The negotfve integers are the "minus" natural numbers: -1, -2. -3, -4, . The rcmonal numbers are all numbers expressible as ~ fractions. The fractions may be proper (less than one; Ex: k) or improper (more than one; Ex: ft). Rational numbers can be positive (Ex: 5.125 = ¥-) or negative (Ex: - ~). All integers are rational: Ex: 4 = y. I SETS A set Is ony collecllon-finite or infinite--<lf things coiled members or elements. To denote a set, we enclose the elements in braces. Ex: N {L,2.3 .... } IS the !infinitel set of natural numbers. The notation (I EN means that a Is in N. or a "is an element of" N. DEFINITIONS -Empty set or null set: 0 or (}: The set without any elements. Beware: the set {O} is a set with one element, O. It is not the same as the em pty set. -Union of two sets: AU B is the set of all elements that are in either set (or in both). Ex: If A = (1,2,3) and B = (2,4,6), thenAUB= (l,2,3,4,6). PROPERTIES OF ARITHMETIC OPERATIONS PROPERTIES OF REAL NUMBERS UNDER ADDITION AND MULTIPLICATION Real numbers satisfy 11 properties: 5 for addition, 5 matching ones for multiplication, and 1 that connects addition and multiplication. Suppose a l b, and c are real numbers. Property Addition (+) Multiplicotion (x or .) Commutotive a + b = b + a a·b=b·a. Assoclotive (a + b) + c = a + (b + c) (b· c) = (a· b) . c ldenlities o is a real number. 1 is a real number. exist a+O=O+a=a a-l=l-a=a o is the additive 1 is the multiplkotive identity. identity. Inverses -a is a real number. If a i- 0, ~ is a real exist a+ (-a) number. =(-a)+a=O u·~=~·a=l Also, -(-a) = a. Also, t = a. Closure a + b is a real number. a b is a real number. LINEAR EQUATIONS IN ONE VARIABLE A linear equation in one variable is an equation that, after simplifying and collecting like terms on each side. will look iike a7 + b = c or like IU" b ("J" + d. Each Side can Involve xs added to real numbers and muihplied by real numbers but not multiplied by other rS Ex: 1(-~-3)+.c=9-(7-~) is a Iineor equation But j" 9 = 3 and 7(X + 4) = 2 and Vi = 5 ore not linear Linear eCJlUlh'ons in one vW'iable will always have (a) exactly one real number solution, (b) rlO solutions, or (c) all real numbers as solutions. FINDING A UNIQUE SOLUTION Ex: J(-t -3) TT=9-(x- ~). 1. Get rid of fractions outside parentheses. Multiply through by the LCM of the denominators. Ex: Multiplyby4 toget3(-t -3) +4x=36-4(x- ~). 2. Simplify using order of operations (PEMDAS). Use the distributive property and combine like terms on each side. Remember to distribute minus signs.
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