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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
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 anequation 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
1.5 Linear Equation in One Variable • Linear Equation in One Variable • Addition and Multiplication Properties of Equality • If
1.5 Solve a Linear Equation • Example Solve Check
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 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
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
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 • Contradiction – 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 • Identity – 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 • Properties of Inequality • 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 • Two Methods • 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. 10 -10 10 10 -15
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.
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
1.5 Solving a Three-Part Inequality • Example Graphical Solution 25 25 -20 6 -20 6 -20 -20