Slide P- 1 Chapter P Prerequisites 1
P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression ( ) 4.5 3. Round the value to two decimal places..3 4.5 4. Evaluate the algebraic expression for the given values 3 of the variable. 1, 1,1.5 x + x x = 5. List the possible remainders when the positive integer n is divided by 6. Slide P- 4
( ) Quick Review Solutions 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression { 1,,3} {-3,-,-1} 4.5 3. Round the value to two deci mal places..73.3 4. 5 4. Evaluate the algebraic expression for the given values 3 of the variable. x + x 1, x = 1,1.5 {-4, 5.375} 5. List the possible remainders when the positive integer n is divided by 6. 1,, 3,4,5 Slide P- 5 What you ll learn about Representing Real Numbers Order and Interval Notation Basic Properties of Algebra Integer Exponents Scientific Notation and why These topics are fundamental in the study of mathematics and science. Slide P- 6 3
Real Numbers A real number is any number that can be written as a decimal. Subsets of the real numbers include: The natural (or counting) numbers: {1,,3 } The whole numbers: {0,1,, } The integers: {,-3,-,-1,0,1,,3, } Slide P- 7 Rational Numbers Rational numbers can be represented as a ratio a/b where a and b are integers and b 0. The decimal form of a rational number either terminates or is indefinitely repeating. Slide P- 8 4
The Real Number Line Slide P- 9 Order of Real Numbers Let a and b be any two real numbers. Symbol Definition Read a>b a b is positive a is greater than b a<b a b is negative a is less than b a b a b is positive or zero a is greater than or equal to b a b a b is negative or zero a is less than or equal to b The symbols >, <,, and are inequality symbols. Slide P- 10 5
Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a < b, a = b, or a > b. Slide P- 11 Example Interpreting Inequalities Describe the graph of x >. Slide P- 1 6
Example Interpreting Inequalities Describe the graph of x >. The inequality describes all real numbers greater than. Slide P- 13 Bounded Intervals of Real Numbers Let a and b be real numbers with a < b. Interval Notation Inequality Notation [a,b] a x b (a,b) a < x < b [a,b) a x< b (a,b] a < x b The numbers a and b are the endpoints of each interval. Slide P- 14 7
Unbounded Intervals of Real Numbers Let a and b be real numbers. Interval Notation Inequality Notation [a, ) x a (a, ) x > a (-,b] x b (-,b) x < b Each of these intervals has exactly one endpoint, namely a or b. Slide P- 15 Properties of Algebra Let u, v, and w be real numbers, variables, or algebraic expressions. 1. Communative Property Addition: u + v = v + u Multiplication uv = vu. Associative Property Addition: ( u + v) + w = u + ( v + w) Multiplication: ( uv) w = u( vw) 3. Identity Property Addition: u + 0 = u Multiplication: u 1 = u Slide P- 16 8
Properties of Algebra Let u, v, and w be real numbers, variables, or algebraic expressions. 4. Inverse Property Addition: u + (- u) = 0 1 Mulitiplication: u = 1, u 0 u 5. Distributive Property Multiplication over addition: u( v + w) = uv + uw ( u + v) w = uw + vw Multiplication over subtraction: u( v w) = uv uw ( u v) w = uw vw Slide P- 17 Properties of the Additive Inverse Let u, v, and w be real numbers, variables, or algebraic expressions. Property Example 1. ( u) = u ( 3) = 3. ( u) v = u( v) = uv ( 4)3 = 4( 3) = 1 3. ( u)( v) = uv ( 6)( 7) = 4 4. ( 1) u = u ( 1)5 = 5 5. ( u + v) = ( u) + ( v) (7 + 9) = ( 7) + ( 9) = 16 Slide P- 18 9
Exponential Notation Let a be a real number, variable, or algebraic expression and n n a positive integer. Then a = a a a... a, where n is the n factors n exponent, a is the base, and a is the nth power of a, read as " a to the nth power." Slide P- 19 Properties of Exponents Let u and v be a real numbers, variables, or algebraic expressions and m and n be integers. All bases are assumed to be nonzero. Property Example m 1. u u m = u + 3 5 5 = 5 = 5 n n 4 3+ 4 7. u u 3. u = 1 8 = 1 0 0 4. u m n x = = = x 9 m n 9 4 5 u x x 4 1 1 = y = n 3 u y - n -3 5. ( uv) m m = u v ( z) = z = 3z m 5 5 5 5 m 6. ( u ) = u ( x ) = x = x n mn 3 3 6 m 7 m 7 u u a a 7. = m = v v b b 7 Slide P- 0 10
Simplify u v 1 u v. 3 Example Simplifying Expressions Involving Powers Slide P- 1 Simplify u v 1 u v. 3 Example Simplifying Expressions Involving Powers u v u u u = = u v v v v 3 1 3 1 3 5 Slide P- 11
Example Converting to Scientific Notation Convert 0.0000345 to scientific notation. Slide P- 3 Example Converting to Scientific Notation Convert 0.0000345 to scientific notation. -5 0.0000345 = 3.45 10 Slide P- 4 1
Example Converting from Scientific Notation Convert 1.3 10 5 from scientific notation. Slide P- 5 Example Converting from Scientific Notation Convert 1.3 10 5 from scientific notation. 13,000 Slide P- 6 13
P. Cartesian Coordinate System Quick Review -5 3 1. Find the distance between and. 4 Use a calculator to evaluate the expression. Round answers to two decimal places.. 8 + 6-1 + 8 3. 4. 3 + 5 ( ) + ( ) 5. 5 1 3 Slide P- 8 14
( ) + ( ) 5 1 3 3. 61 Quick Review Solutions -5 3 1. Find the distance between and..75 4 Use a calculator to evaluate the expression. Round answers to two decimal places.. 8 + 6-1 + 8 3. 4. 3 + 5 5. - 10 5.83 Slide P- 9 What you ll learn about Cartesian Plane Absolute Value of a Real Number Distance Formulas Midpoint Formulas Equations of Circles Applications and why These topics provide the foundation for the material that will be covered in this textbook. Slide P- 30 15
The Cartesian Coordinate Plane Slide P- 31 Quadrants Slide P- 3 16
Absolute Value of a Real Number The absolute value of a real number a a, if a > 0 a = a if a < 0. 0, if a = 0 is Slide P- 33 Properties of Absolute Value Let a and b be real numbers. 1. a 0. - a = a 3. ab = a b a a 4. =, b 0 b b Slide P- 34 17
Distance Formula (Number Line) Let a and b be real numbers. The distance between a and b is a b. Note that a b = b a. Slide P- 35 Distance Formula (Coordinate Plane) The distance d between points P(x, y ) Q(x, y ) in the and 1 1 ( ) ( ) 1 1 coordinate plane is d = x x + y y. Slide P- 36 18
The Distance Formula using the Pythagorean Theorem Slide P- 37 Midpoint Formula (Number Line) The a + b. midpoint of the line segment with endpoints a and b is Slide P- 38 19
Midpoint Formula (Coordinate Plane) The midpoint of the line segment with endpoints ( a,b) and ( c,d ) a + c b + d,. is Slide P- 39 Standard Form Equation of a Circle The standard form equation of a circle with center ( h, k) and radius r is ( x h) + ( y k) = r. Slide P- 40 0
Standard Form Equation of a Circle Slide P- 41 Example Finding Standard Form Equations of Circles Find the standard form equation of the circle with center (, 3) and radius 4. Slide P- 4 1
Example Finding Standard Form Equations of Circles Find the standard form equation of the circle with center (, 3) and radius 4. ( x h) + ( y k) = r where h =, k = 3,and r = 4. Thus the equation is ( x ) + ( y + 3) = 16. Slide P- 43 P.3 Linear Equations and Inequalities
Quick Review Simplify the expression by combining like terms. 1. x + 4x y y 3x. 3(x ) + 4( y 1) Use the LCD to combine the fractions. Simplify the resulting fraction. 3 4 3. + x x x + x 4. + 4 3 5. + y Slide P- 45 Quick Review Solutions Simplify the expression by combining like terms. 1. x + 4x y y 3 x. 3(x ) + 4( y 1) Use the LCD to combine the fractions. Simplify the resulting fraction. 3 4 7 3. + x x x x + x 7x + 6 4. + 4 3 1 y + 5. + y y 3x 3y 6x + 4y 10 Slide P- 46 3
Equations Solving Equations What you ll learn about Linear Equations in One Variable Linear Inequalities in One Variable and why These topics provide the foundation for algebraic techniques needed throughout this textbook. Slide P- 47 Properties of Equality Let u, v, w, and z be real numbers, variables, or algebraic expressions. 1. Reflexive u = u. Symmetric If u = v, then v = u. 3. Transitive If u = v, and v = w, then u = w. 4. Addition If u = v and w = z, then u + w = v + z. 5. Multiplication If u = v and w = z, then uw = vz. Slide P- 48 4
Linear Equations in x A linear equation in x is one that can be written in the form ax + b = 0, where a and b are real numbers with a 0. Slide P- 49 Operations for Equivalent Equations An equivalent equation is obtained if one or more of the following operations are performed. Operation Given Equation Equivalent Equation 1. Combine like terms, 3 1 x + x = 3x = 9 3 reduce fractions, and remove grouping symbols. Perform the same operation on both sides. (a) Add ( 3) x + 3 = 7 x = 4 ( b) Subtract ( x) 5x = x + 4 3x = 4 (c) Multiply by a nonzero constant (1/3) 3x = 1 x = 4 (d) Divide by a constant nonzero term (3) 3x = 1 x = 4 Slide P- 50 5
Example Solving a Linear Equation Involving Fractions 10y 4 y Solve for y. = + 4 4 Slide P- 51 Example Solving a Linear Equation Involving Fractions 10y 4 y Solve for y. = + 4 4 10y 4 y = + 4 4 10y 4 y 4 = + 4 4 4 Multiply by the LCD 10y 4 = y + 8 Distributive Property 9y = 1 Simplify 4 y = 3 Slide P- 5 6
Linear Inequality in x A linear inequality in x is one that can be written in the form ax + b < 0, ax + b 0, ax + b > 0, or ax + b 0, where a and b are real numbers with a 0. Slide P- 53 Properties of Inequalities Let u, v, w, and z be real numbers, variables, or algebraic expressions, and c a real number. 1. Transitive If u < v, and v < w, then u < w.. Addition If u < v then u + w < v + w. If u < v and w < z then u + w < v + z. 3. Multiplication If u < v and c > 0, then uc < vc. If u < v and c < 0, then uc > vc. The above properties are true if < is replaced by. There are similar properties for > and. Slide P- 54 7
P.4 Lines in the Plane Quick Review Solve for x. 1. 50x + 100 = 00. 3(1 x) + 4( x + ) = 10 Solve for y. 3. x 3y = 5 4. x 3( x + y) = y 5. Simplify the fraction. 7 10 ( 3) Slide P- 56 8
Quick Review Solutions Solve for x. 1. 50x + 100 = 00. 3(1 x) + 4( x + ) = 10 Solve for y. x = x = 1 x 5 3. x 3y = 5 y = 3 x 4. x 3( x + y) = y y = 4 7 5. Simplify the fraction. 10 ( 3) 5 7 Slide P- 57 What you ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables Parallel and Perpendicular Lines Applying Linear Equations in Two Variables and why Linear equations are used extensively in applications involving business and behavioral science. Slide P- 58 9
Slope of a Line Slide P- 59 Slope of a Line The slope of the nonvertical line through the points ( x, y ) y y y x x x 1 and ( x, y ) is m = =. 1 If the line is vertical, then x = x 1 1 1 and the slope is undefined. Slide P- 60 30
Example Finding the Slope of a Line Find the slope of the line containing the points (3,-) and (0,1). Slide P- 61 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-) and (0,1). y y 1 ( ) 3 m x x 0 3 3 1 = = = = 1 Thus, the slope of the line is 1. 1 Slide P- 6 31
Point-Slope Form of an Equation of a Line The point - slope form of an equation of a line that passes through the point ( x, y ) and has slope m is y y = m( x x ). 1 1 1 1 Slide P- 63 Point-Slope Form of an Equation of a Line Slide P- 64 3
Slope-Intercept Form of an Equation of a Line The slope-intercept form of an equation of a line with slope m and y-intercept (0,b) is y = mx + b. Slide P- 65 Forms of Equations of Lines General form: Ax + By + C = 0, A and B not both zero Slope-intercept form: y = mx + b Point-slope form: y y 1 = m(x x 1 ) Vertical line: x = a Horizontal line: y = b Slide P- 66 33
Graphing with a Graphing Utility To draw a graph of an equation using a grapher: 1. Rewrite the equation in the form y = (an expression in x).. Enter the equation into the grapher. 3. Select an appropriate viewing window. 4. Press the graph key. Slide P- 67 Viewing Window Slide P- 68 34
Parallel and Perpendicular Lines 1. Two nonvertical lines are parallel if and only if their slopes are equal.. Two nonvertical lines are perpendicular if and only if their slopes m and m are opposite reciprocals. 1 1 That is, if and only if m =. 1 m Slide P- 69 Example Finding an Equation of a Parallel Line Find an equation of a line through (, 3) that is parallel to 4 x + 5 y = 10. Slide P- 70 35
Example Finding an Equation of a Parallel Line Find an equation of a line through (, 3) that is parallel to 4 x + 5 y = 10. Find the slope of 4 x + 5 y = 10. 5y = 4x + 10 4 4 y = x + The slope of this line is. 5 5 Use point-slope form: 4 y + 3 = 5 ( x ) Slide P- 71 P.5 Solving Equations Graphically, Numerically, and Algebraically 36
Expand the product. ( x + y) ( x + )( x ) 1.. 1 4 3 Factor completely. 3. 3 x + x x 4. 5 36 4 y + y Quick Review 5. Combine the fractions and reduce the resulting fraction x to lowest terms. x + 1 x 1 Slide P- 73 Quick Review Solutions Expand the product. ( x + y) ( x + )( x ) 1.. 1 4 3 Factor completely. 3. x + x x 3 4. y + 5y 36 x + 4xy + 4y 4 8x x 3 ( x + 1)( x 1)( x + ) ( y + 9)( y )( y + ) 5. Combine the fractions and reduce the resulting fraction to lowest terms. x x 5x + x + 1 x 1 1 ( x + 1)( x ) Slide P- 74 37
What you ll learn about Solving Equations Graphically Solving Quadratic Equations Approximating Solutions of Equations Graphically Approximating Solutions of Equations Numerically with Tables Solving Equations by Finding Intersections and why These basic techniques are involved in using a graphing utility to solve equations in this textbook. Slide P- 75 Example Solving by Finding x-intercepts Solve the equation x 3x = 0 graphically. Slide P- 76 38
Example Solving by Finding x-intercepts Solve the equation x 3x = 0 graphically. Find the x-intercepts of y = x 3x. Use the Trace to see that ( 0.5,0) and (,0) are x-intercepts. Thus the solutions are x = 0.5 and x =. Slide P- 77 Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0. Slide P- 78 39
Quadratic Equation in x A quadratic equation in x is one that can be written in the form ax + bx + c = 0, where a, b, and c are real numbers with a 0. Slide P- 79 Completing the Square To solve x + bx = c by completing the square, add ( b / ) to both sides of the equation and factor the left side of the new equation. b b x + bx + = c + b b x + = c + 4 Slide P- 80 40
Quadratic Equation The solutions of the quadratic equation a 0, are given by the quadratic formula ± x = b b ac 4 a. ax bx c + + = 0, where Slide P- 81 Example Solving Using the Quadratic Formula Solve the equation x + 3x 5 = 0. Slide P- 8 41
Example Solving Using the Quadratic Formula Solve the equation x + 3x 5 = 0. a =, b = 3, c = 5 ± x = b b 4ac a ( )( ) ( ) ± = 3 3 4 5 3 ± 49 = 4 3 ± 7 = 4 x = 5 or x = 1. Slide P- 83 Solving Quadratic Equations Algebraically These are four basic ways to solve quadratic equations algebraically. 1. Factoring. Extracting Square Roots 3. Completing the Square 4. Using the Quadratic Formula Slide P- 84 4
Agreement about Approximate Solutions For applications, round to a value that is reasonable for the context of the problem. For all others round to two decimal places unless directed otherwise. Slide P- 85 Example Solving by Finding Intersections Solve the equation x 1 = 6. Slide P- 86 43
Example Solving by Finding Intersections Solve the equation x 1 = 6. Graph y = x 1 and y = 6. Use Trace or the intersect feature of your grapher to find the points of intersection. The graph indicates that the solutions are x =.5 and x = 3.5. Slide P- 87 P.6 Complex Numbers 44
Quick Review Add or subtract, and simplify. 1. (x + 3) + ( x + 3). (4x 3) ( x + 4) Multiply and simplify. 3. ( x + 3)( x ) ( x + )( x ) 4. 3 3 5. (x + 1)(3x + 5) Slide P- 89 Add or subtract, and simplify. 1. (x + 3) + ( x + 3). (4x 3) ( x + 4) Multiply and simplify. 3. ( x + 3)( x ) ( x + )( x ) 4. 3 3 5. (x + 1)( 3 x + ) + Quick Review Solutions x + 6 3x 7 x + x 6 x 3 5 6x + 13x 5 Slide P- 90 45
What you ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations and why The zeros of polynomials are complex numbers. Slide P- 91 Complex Number A complex number is any number that can be written in the form a + bi, where a and b are real numbers. The real number a is the real part, the real number b is the imaginary part, and a + bi is the standard form. Slide P- 9 46
Addition and Subtraction of Complex Numbers If a + bi and c + di are two complex numbers, then Sum: (a + bi ) + (c + di ) = (a + c) + (b + d)i, Difference: (a + bi ) (c + di ) = (a - c) + (b -d)i. Slide P- 93 Example Multiplying Complex Numbers ( + i)( i) Find 3 4. Slide P- 94 47
Example Multiplying Complex Numbers ( + i)( i) Find 3 4. ( 3 + i)( 4 i) = 1 3i + 8i i = 1 + 5i ( 1) = 1 + 5i + = 14 + 5i Slide P- 95 Complex Conjugate The complex conjugate of the complex number z = a + bi is z = a + bi = a bi. Slide P- 96 48
Discriminant of a Quadratic Equation For a quadratic equation + + = 0, where,, and are real numbers and a 0. i i i if b 4ac > 0, there are two distinct real solutions. if b 4ac = 0, there is one repeated real solution. if b 4ac < 0, the ax bx c a b c re is a complex pair of solutions. Slide P- 97 Example Solving a Quadratic Equation Solve x + x + = 0. Slide P- 98 49
Example Solving a Quadratic Equation Solve x + x + = 0. a = b = 1, and c =. ± x = = = ( ) ( ) 1 1 4 1 1 1± 7 1± i 7 1+ i 7 1 i 7 So the solutions are x = and x =. Slide P- 99 P.7 Solving Inequalities Algebraically and Graphically 50
Quick Review Solve for x. 1. 3 < x + 1 < 9. x + 1 = 3 3. Factor completely. 4x 9 x 49 4. Reduce the fraction to lowest terms. x + 7x x x + 5. Add the fractions and simplify. + x + 1 x Slide P- 101 Solve for x. 1. 3 < x + 1 < 9 < x < 4. x + 1 = 3 x = or x = 1 3. Factor completely. 4x 9 Quick Review Solutions ( x 3)( x 3) + x 49 x 7 4. Reduce the fraction to lowest terms. x + 7x x x x + x + 3x + 5. Add the fractions and simplify. + x + 1 x x + x Slide P- 10 51
What you ll learn about Solving Absolute Value Inequalities Solving Quadratic Inequalities Approximating Solutions to Inequalities Projectile Motion and why These techniques are involved in using a graphing utility to solve inequalities in this textbook. Slide P- 103 Solving Absolute Value Inequalities Let u be an algebraic expression in x and let a be a real number with a 0. 1. If u < a, then u is in the interval ( a, a). That is, u < a if and only if a < u < a.. If u > a, then u is in the interval (, a) or ( a, ). That is, u > a if and only if u < a or u > a. The inequalities < and > can be replaced with and, respectively. Slide P- 104 5
Solve x + 3 < 5. Example Solving an Absolute Value Inequality Slide P- 105 Solve x + 3 < 5. Example Solving an Absolute Value Inequality x + 3 < 5 5 < x + 3 < 5 8 < x < As an interval the solution in ( 8, ). Slide P- 106 53
Example Solving a Quadratic Inequality Solve x + 3x + < 0. Slide P- 107 Example Solving a Quadratic Inequality Solve x + 3x + < 0. Solve x + 3x + = 0 ( x + )( x + 1) = 0 x = or x = 1. Use these solutions and a sketch of the equation y x x = + 3 + to find the solution to the inequality in interval form (, 1). Slide P- 108 54
Projectile Motion Suppose an object is launched vertically from a point s o feet above the ground with an initial velocity of v o feet per second. The vertical position s (in feet) of the object t seconds after it is launched is s = -16t + v o t + s o. Slide P- 109 Chapter Test 1. Write the number in scientific notation. The diameter of a red blood corpuscle is about 0.000007 meter.. Find the standard form equation for the circle with center (5, 3) and radius 4. 3. Find the slope of the line through the points ( 1, ) and (4, 5). 4. Find the equation of the line through (, 3) and perpendicular to the line x + 5y = 3. x x + 5 1 5. Solve the equation algebraically. + = 3 3 6. Solve the equation algebraically. 6x + 7x = 3 Slide P- 110 55
Chapter Test 7. Solve the equation algebraically. 4x + 1 = 3 8. Solve the inequality. 3x + 4 9. Solve the inequality. 4x + 1x + 9 0 10. Perform the indicated operation, and write the result in standard form. (5 7 i) (3 i) Slide P- 111 56