Transcript sec 1.5
Copyright © 2007 Pearson Education, Inc. Slide 1-1 Chapter 1: Linear Functions, Equations, and Inequalities 1.1 Real Numbers and the Rectangular Coordinate System 1.2 Introduction to Relations and Functions 1.3 Linear Functions 1.4 Equations of Lines and Linear Models 1.5 Linear Equations and Inequalities 1.6 Applications of Linear Functions Copyright © 2007 Pearson Education, Inc. Slide 1-2 1.5 Linear Equations and Inequalities • Solving Linear Equations – analytic: paper & pencil – graphic: often supports analytic approach with graphs and tables • Equations – statements that two expressions are equal – to solve an equation means to find all numbers that will satisfy the equation – the solution to an equation is said to satisfy the equation – solution set is the list of all solutions Copyright © 2007 Pearson Education, Inc. Slide 1-3 1.5 Linear Equation in One Variable • Linear Equation in One Variable ax b 0, a 0 • Addition and Multiplication Properties of Equality – a b and a c b c are equivalent – If c 0, then a b and ac bc are equivalent Copyright © 2007 Pearson Education, Inc. Slide 1-4 1.5 Solve a Linear Equation • Example Solve 10 3(2 x 4) 17 ( x 5) 10 6 x 12 17 x 5 2 7 x 12 7 x 14 x2 Distributi ve property Add x to each side Add 2 to each side Divide each side by 7 Check 10 3(2(2) 4) 17 (2 5) 10 0 17 7 10 10 Copyright © 2007 Pearson Education, Inc. Slide 1-5 1.5 Solve a Linear Equation with Fractions x 7 2x 8 4 • Solve 6 2 x 7 2 x 8 6 6(4) 2 6 x 7 3(2 x 8) 24 x 7 6 x 24 24 7 x 17 24 7 x 7 x 1 Copyright © 2007 Pearson Education, Inc. Slide 1-6 1.5 Graphical Solutions to f (x) = g(x) • Three possible solutions y y x 1 point Copyright © 2007 Pearson Education, Inc. y x No points x Infinitely many points (coincide) Slide 1-7 1.5 Intersection-of-Graphs Method • First Graphical Approach to Solving Linear Equations – f ( x) g ( x), where f and g are linear functions 1. set y1 f ( x) and y2 g ( x) and graph 2. find points of intersection, if any, using intersect in the CALC menu – e.g. 10 3(2 x 4) 17 ( x 5) Copyright © 2007 Pearson Education, Inc. Slide 1-8 1.5 Application • The percent share of music sales (in dollars) that compact discs (CDs) held from 1987 to 1998 can be modeled by f ( x ) 5.91x 13.7. During the same time period, the percent share of music sales that cassette tapes held can be modeled by g ( x ) 4.71x 64.7. In these formulas, x = 0 corresponds to 1987, x = 1 to 1988, and so on. Use the intersection-of-graphs method to estimate the year when sales of CDs equaled sales of cassettes. Solution: 4.71x 64.7 5.91x 13.7 100 1987 4.8 1992 . It follows that in 1992, both CDs and cassettes shared about 42.1% of sales. 0 Copyright © 2007 Pearson Education, Inc. 12 Slide 1-9 1.5 The x-Intercept Method • Second Graphical Approach to Solving a Linear Equation f ( x) g ( x) f ( x) g ( x) 0 – set y1 f ( x) g ( x) and any x-intercept (or zero) is a solution of the equation • Root, solution, and zero refer to the same basic concept: – real solutions of f ( x) 0 correspond to the x-intercepts of the graph y f (x) Copyright © 2007 Pearson Education, Inc. Slide 1-10 1.5 Example Using the x-Intercept Method • Solve the equation 6 x 4(3 2 x) 5( x 4) 10 6 x 4(3 2 x) (5( x 4) 10) 0 Graph hits x-axis at x = –2. Use Zero in CALC menu. Copyright © 2007 Pearson Education, Inc. Slide 1-11 1.5 Identities and Contradictions • Contradiction – equation that has no solution – e.g. x x 3 y2 x 3 y1 x two parallel lines The solution set is the empty or null set, denoted . Copyright © 2007 Pearson Education, Inc. Slide 1-12 1.5 Identities and Contradictions • Identity – equation that is true for all values in the domain – e.g. 2( x 3) 2 x 6 lines coincide Solution set (, ). Copyright © 2007 Pearson Education, Inc. Slide 1-13 1.5 Identities and Contradictions • Note: – Contradictions and identities are not linear, since linear equations must be of the form ax b 0, a 0 – linear equations - one solution – contradictions - always false – identities - always true Copyright © 2007 Pearson Education, Inc. Slide 1-14 1.5 Solving Linear Inequalities • Properties of Inequality a. a b and a c b c are equivalent b. If c 0, then a b and ac bc are equivalent c. If c 0, then a b and ac bc are equivalent • Example 3 x 2(2 x 4) 2 x 1 3x 4 x 8 2 x 1 x 8 2x 1 3x 9 x 3 or [3,) Copyright © 2007 Pearson Education, Inc. Slide 1-15 1.5 Solve a Linear Inequality with Fractions 2x 3 x2 3 6x 9 x 2 Reverse the inequality symbol when multiplying by a negative number. 9 7x 2 7 7x 1 x or x 1 The solution set is (,1). Copyright © 2007 Pearson Education, Inc. Slide 1-16 1.5 Graphical Approach to Solving Linear Inequalities • Two Methods 1. Intersection-of-Graphs f ( x) g ( x) where the solution is the set of all real numbers x such that f is above the graph of g. – Similarly for f ( x ) g ( x ), f is below the graph of g. – e.g. 3 x 2( 2 x 4) 2 x 1 – 10 -10 10 y2 2 x 1 10 y1 3 x 2( 2 x 4) -15 Solution set : [ 3, ) Copyright © 2007 Pearson Education, Inc. Slide 1-17 1.5 Graphical Approach to Solving Linear Inequalities 2. x-intercept Method – F ( x ) 0 is the set of all real numbers x such that the graph of F is above the x-axis. – Similarly for F (x) < 0, the graph of F is below the x-axis. – e.g. 2(3 x 1) 4( x 2) Let y1 2 (3 x 1) and y 2 4 ( x 2 ), then y1 y 2 can be written as y1 y 2 0. So we graph y1 y 2 2 (3 x 1) 4 ( x 2 ) 0 and find the x - intercept using the zero function in the CALC menu. y1 2 ( 3 x 1) 4 ( x 2 ) Solution set : (-1, ) Copyright © 2007 Pearson Education, Inc. Slide 1-18 1.5 Three-Part Inequalities • Application – error tolerances in manufacturing a can with radius of 1.4 inches • r can vary by 0.02 inches 1.38 r 1.42 • Circumference C 2r varies between 2 (1.38) 8.67 inches and 2 (1.42) 8.92 inches 8.67 C 8.92 r Copyright © 2007 Pearson Education, Inc. Slide 1-19 1.5 Solving a Three-Part Inequality • Example 2 5 3x 20 7 3x 15 7 x5 3 The solution set is ( -37 ,5) Graphical Solution y 3 20 y 3 20 25 25 y2 5 3x y2 5 3x -20 -20 Copyright © 2007 Pearson Education, Inc. -20 6 y1 2 6 -20 y1 2 Slide 1-20