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Chapter 1: Linear Functions, Equations, and Inequalities

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

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Chapter 1: Linear Functions, Equations, and Inequalities

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  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

  2. 1.5 Linear Equations and Inequalities • Equations • statements that two expressions are equal • to solve anequation means to find all numbers that will satisfy the equation • the solution (or root) of an equation is said to satisfy the equation • solution set is the list of all solutions

  3. 1.5 Linear Equation in One Variable A linear equation in one variable is an equation that can be written in the form

  4. 1.5 Linear Equations and Inequalities • Solving Linear Equations • analytic: paper & pencil • graphical: often supports analytic approach with graphs and tables

  5. 1.5 Addition and Multiplication Properties • Addition and Multiplication Properties of Equality For real numbers a, b, and c,

  6. 1.5 Solving a Linear Equation • Example Solve Check

  7. 1.5 Solving a Linear Equation with Fractions • Solve

  8. 1.5 Graphical Solutions to f(x) = g(x) • Three possible solutions

  9. 1.5 Intersection-of-Graphs Method • First Graphical Approach to Solving Linear Equations • where f and g are linear functions • set and graph • find points of intersection, if any, using intersect in the CALC menu • e.g.

  10. 1.5 Intersection-of-Graphs Method Intersection-of-Graphs Method of Graphical Solution To solve the equation graphically, solve The x-coordinate of any point of intersection of the two graphs is a solution of the equation.

  11. 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: 100 0 12

  12. 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

  13. 1.5 The x-Intercept Method x-intercept Method of Graphical Solution To solve the equation graphically, solve The x-intercept of the graph of F (or zero of the function F) is a solution of the equation.

  14. 1.5 The x-Intercept Method • Root, solution, and zero refer to the same basic concept: • real solutions of correspond to the x-intercepts of the graph

  15. 1.5 Example Using the x-Intercept Method • Solve the equation Graph hits x-axis at x = –2. Use Zero in CALC menu.

  16. 1.5 Identities and Contradictions • A contradiction is an equation that has no solution. • e.g. The solution set is the empty or null set, denoted two parallel lines

  17. 1.5 Identities and Contradictions • An identity is an equation that is true for all values in the domain. • e.g. Solution set lines coincide

  18. 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

  19. 1.5 Solving Linear Inequalities Addition and Multiplication Properties of Inequality

  20. 1.5 Solving Linear Inequalities • Example

  21. 1.5 Solve a Linear Inequality with Fractions Reverse the inequality symbol when multiplying by a negative number.

  22. 1.5 Graphical Approach to Solving Linear Inequalities Intersection-of-Graphs Method of Solution of a Linear Inequality Suppose thatf and g are linear functions. The solution set of is the set of all real numbers x such that the graph of f is above the graph of g. The solution set of is the set of all real numbers x such that the graph of f is below the graph of g.

  23. 1.5 Intersection of Graphs Method Example: 10 -10 10 10 -15

  24. 1.5 Intersection of Graphs Method Agreement of Inclusion of Exclusion of Endpoints for Approximations When an approximation is used for an endpoint in specifying an interval, we continue to use parentheses in specifying inequalities involving < or > and square brackets in specifying inequalities involving < or >.

  25. 1.5 x-Intercept Method x-intercept Method of Solution of a Linear Inequality The solution set of is the set of all real numbers x such that the graph of F is above the x-axis. The solution set of is the set of all real numbers x such that the graph of F is below the x-axis.

  26. 1.5 x-Intercept Method Example:

  27. 1.5 Three-Part Inequalities • Application Consider error tolerances in manufacturing a can with radius of 1.4 inches. • r can vary by • Circumference varies between and r

  28. 1.5 Solving a Three-Part Inequality • Example Graphical Solution 25 25 -20 6 -20 6 -20 -20

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