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

2. 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

3. Copyright © 2007 Pearson Education, Inc. Slide 1-3 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

4. Copyright © 2007 Pearson Education, Inc. Slide 1-4 1.5 Linear Equation in One Variable Linear Equation in One Variable Addition and Multiplication Properties of Equality – – If

5. Copyright © 2007 Pearson Education, Inc. Slide 1-5 1.5 Solve a Linear Equation Example Solve Check

6. Copyright © 2007 Pearson Education, Inc. Slide 1-6 1.5 Solve a Linear Equation with Fractions Solve

7. Copyright © 2007 Pearson Education, Inc. Slide 1-7 1.5 Graphical Solutions to f (x) = g(x) Three possible solutions

8. Copyright © 2007 Pearson Education, Inc. Slide 1-8 1.5Intersection-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.

9. Copyright © 2007 Pearson Education, Inc. Slide 1-9 1.5 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: 012 100

10. Copyright © 2007 Pearson Education, Inc. Slide 1-10 1.5 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

11. Copyright © 2007 Pearson Education, Inc. Slide 1-11 1.5 Example Using the x-Intercept Method Solve the equation Graph hits x-axis at x = –2. Use Zero in CALC menu.

12. 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

13. Copyright © 2007 Pearson Education, Inc. Slide 1-13 1.5 Identities and Contradictions Identity – equation that is true for all values in the domain –e.g. Solution set lines coincide

14. Copyright © 2007 Pearson Education, Inc. Slide 1-14 1.5 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

15. Copyright © 2007 Pearson Education, Inc. Slide 1-15 1.5 Solving Linear Inequalities Properties of Inequality a. b. c. Example

16. Copyright © 2007 Pearson Education, Inc. Slide 1-16 1.5 Solve a Linear Inequality with Fractions Reverse the inequality symbol when multiplying by a negative number.

17. Copyright © 2007 Pearson Education, Inc. Slide 1-17 1.5 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. -1010 -15

18. Copyright © 2007 Pearson Education, Inc. Slide 1-18 1.5 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.

19. Copyright © 2007 Pearson Education, Inc. Slide 1-19 1.5 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

20. Copyright © 2007 Pearson Education, Inc. Slide 1-20 1.5 Solving a Three-Part Inequality Example Graphical Solution -20 25 -20 6 6