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CHAPTER 1 REAL NUMBERS AND THEIR BASIC PROPERTIES Natural numbers: {1, 2, 3, 4, 5, . . .} Whole numbers: {0, 1, 2, 3, 4, 5, . . .} Integers: {. . . , 3, 2, 1, 0, 1, 2, 3, . . .} Rational numbers: {All numbers that can be written as a fraction with an integer numerator and a nonzero integer denominator} Real numbers: {All numbers that are either a rational number or an irrational number} Prime numbers: {2, 3, 5, 7, 11, 13, 17, . . .} Composite numbers: {4, 6, 8, 9, 10, 12, 14, 15, . . .} Even integers: {. . . , 6, 4, 2, 0, 2, 4, 6, . . .} Odd integers: {. . . , 5, 3, 1, 1, 3, 5, . . .} Fractions: If there are no divisions by 0, then ax a a c ac bx b b d bd a c ad a b ab b d bc d d d a b ab d d d Exponents and order of operations: If n is a natural number, then n factors of x
x xxx p x To simplify expressions, do all calculations within each pair of grouping symbols, working from the innermost pair to the outermost pair. 1. Find the values of any exponential expressions. 2. Do all multiplications and divisions from left to right. 3. Do all additions and subtractions from left to right. In a fraction, simplify the numerator and denominator separately and then simplify the fraction, if possible. n
Figure
Perimeter
Area
Square Rectangle
P 4s P 2l 2w
Triangle
Pabc
Trapezoid
Pabcd
Circle
C pD 2pr
A s2 A lw 1 A bh 2 1 A h(b d) 2 A pr2
Figure
Volume
Rectangular solid Cylinder
V lwh V Bh* 1 V Bh* 3 1 V Bh* 3 4 3 V pr 3
Pyramid Cone Sphere *B is the area of the base.
If a, b, and c are real numbers, then Closure properties: a b is a real number. a b is a real number. ab is a real number. a is a real number (b 0). b Commutative properties: abba ab ba Associative properties: (a b) c a (b c) (ab)c a(bc) Distributive property: a(b c) ab ac CHAPTER 2 EQUATIONS AND INEQUALITIES Let a, b, and c be real numbers. If a b, then a c b c. If a b, then a c b c. b a If a b, then (c 0). c c If a b, then ca cb. Sale price regular price markdown Retail price wholesale cost markup Percentage rate base Solving inequalities: Let a, b, and c be real numbers. If a b, then a c b c. If a b, then a c b c. If a b and c 0, then ac bc.
If a b and c 0, then ac bc. a b If a b and c 0, then . c c If a b and c 0, then
a b . c c
CHAPTER 3 GRAPHING AND SOLVING SYSTEMS OF EQUATIONS AND INEQUALITIES General form of the equation of a line: Ax By C Equation of a vertical line: x a Equation of a horizontal line: y b CHAPTER 4 POLYNOMIALS Properties of exponents: If x is a number, m and n are integers, and there are no divisions by 0, then n factors of x
x xxx p x xmxn xmn (xm)n xmn x n xn (xy)n xnyn a b n y y n
xm xmn xn 1 xn n x
x0 1 xm xmn xn
Special products: (x y)2 x2 2xy y2 (x y)2 x2 2xy y2 (x y)(x y) x2 y2 CHAPTER 5 FACTORING POLYNOMIALS Factoring the difference of two squares: a2 b2 (a b)(a b) Factoring perfect-square trinomials: a2 2ab b2 (a b)2 a2 2ab b2 (a b)2 Factoring the sum and difference of two cubes: x3 y3 (x y)(x2 xy y2) x3 y3 (x y)(x2 xy y2) Zero-factor property: Let a and b be real numbers. If ab 0, then a 0 or b 0.
CHAPTER 6 PROPORTIONS AND RATIONAL EXPRESSIONS a c , then ad bc. b d If there are no divisions by 0, then If
ac a a a a is undefined bc b 1 0 a c ad a b ab b d bc d d d a b ab d d d
a c ac b d bd
CHAPTER 7 MORE EQUATIONS, INEQUALITIES, AND FACTORING If x 0,