1.5 Linear Equations and Inequalities statements that two expressions are equal to solve an equation 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

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

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

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

1.5 Solving a Linear Equation Example Solve Check

1.5 Solving a Linear Equation with Fractions Solve

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

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.

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.

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 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

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.

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

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

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

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

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

1.5 Solving Linear Inequalities Addition and Multiplication Properties of Inequality

1.5 Solving Linear Inequalities Example

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

1.5 Graphical Approach to Solving Linear Inequalities Intersection-of-Graphs Method of Solution of a Linear Inequality Suppose that f 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.

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

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 >.

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.

1.5 x-Intercept Method Example:

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

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