Copyright © 2007 Pearson Education, Inc. Slide 1-1

Copyright © 2007 Pearson Education, Inc. Slide 1-2 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 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 Linear Equation in One Variable Linear Equation in One Variable Addition and Multiplication Properties of Equality – – If

Copyright © 2007 Pearson Education, Inc. Slide Solve a Linear Equation Example Solve Check

Copyright © 2007 Pearson Education, Inc. Slide Solve a Linear Equation with Fractions Solve

Copyright © 2007 Pearson Education, Inc. Slide Graphical Solutions to f (x) = g(x) Three possible solutions

Copyright © 2007 Pearson Education, Inc. Slide Intersection-of-Graphs Method First Graphical Approach to Solving Linear Equations – where f and g are linear functions 1.set and graph 2.find points of intersection, if any, using intersect in the CALC menu –e.g.

Copyright © 2007 Pearson Education, Inc. Slide Application The percent share of music sales (in dollars) that compact discs (CDs) held from 1987 to 1998 can be modeled by During the same time period, the percent share of music sales that cassette tapes held can be modeled by 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:

Copyright © 2007 Pearson Education, Inc. Slide The x-Intercept Method Second Graphical Approach to Solving a Linear Equation –set 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 correspond to the x-intercepts of the graph

Copyright © 2007 Pearson Education, Inc. Slide Example Using the x-Intercept Method Solve the equation Graph hits x-axis at x = –2. Use Zero in CALC menu.

Copyright © 2007 Pearson Education, Inc. Slide 1-12 two parallel lines 1.5 Identities and Contradictions Contradiction – equation that has no solution –e.g. The solution set is the empty or null set, denoted

Copyright © 2007 Pearson Education, Inc. Slide Identities and Contradictions Identity – equation that is true for all values in the domain –e.g. Solution set lines coincide

Copyright © 2007 Pearson Education, Inc. Slide Identities and Contradictions Note: –Contradictions and identities are not linear, since linear equations must be of the form –linear equations - one solution –contradictions - always false –identities - always true

Copyright © 2007 Pearson Education, Inc. Slide Solving Linear Inequalities Properties of Inequality a. b. c. Example

Copyright © 2007 Pearson Education, Inc. Slide Solve a Linear Inequality with Fractions Reverse the inequality symbol when multiplying by a negative number.

Copyright © 2007 Pearson Education, Inc. Slide Graphical Approach to Solving Linear Inequalities Two Methods 1.Intersection-of-Graphs – where the solution is the set of all real numbers x such that f is above the graph of g. –Similarly for f is below the graph of g. –e.g

Copyright © 2007 Pearson Education, Inc. Slide Graphical Approach to Solving Linear Inequalities 2. x-intercept Method – 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.

Copyright © 2007 Pearson Education, Inc. Slide Three-Part Inequalities Application –error tolerances in manufacturing a can with radius of 1.4 inches r can vary by Circumference varies between and r

Copyright © 2007 Pearson Education, Inc. Slide Solving a Three-Part Inequality Example Graphical Solution