1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 1 Equations and Inequalities

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Linear Equations in One Variable Learn the vocabulary and concepts used in studying equations. Solve linear equations in one variable. Solve rational equations with variables in the denominators. Solve formulas for a specific variable. Solve applied problems by using linear equations. SECTION

Definitions An equation is a statement that two mathematical expressions are equal. For example, An equation in one variable, is a statement that two expressions, with at least one containing the variable, are equal. For example, The domain of the variable in an equation is the set of all real numbers for which both sides of the equation are defined. © 2010 Pearson Education, Inc. All rights reserved 3 3

EXAMPLE 1a Finding the Domain of the Variable Find the domain of the variable x in each of the following equations. Solution Left side is defined for all values of x. Right side is not defined if x = 1. Simply write © 2010 Pearson Education, Inc. All rights reserved 4

EXAMPLE 1b Finding the Domain of the Variable Find the domain of the variable x in each of the following equations. Solution Right side is defined for x ≥ 0. Left side is is defined for all values of x. The domain is Interval notation 5 © 2010 Pearson Education, Inc. All rights reserved 5

EXAMPLE 1c Finding the Domain of the Variable Find the domain of the variable x in each of the following equations. Solution Both sides are defined for all values of x. The domain is Interval notation © 2010 Pearson Education, Inc. All rights reserved 6

LINEAR EQUATIONS A conditional linear equation in one variable, such as x, is an equation that can be written in the standard form where a and b are real numbers with a ≠ 0. © 2010 Pearson Education, Inc. All rights reserved 7

PROCEDURE FOR SOLVING LINEAR EQUATIONS IN ONE VARIABLE Step 1Eliminate Fractions. Multiply both sides of the equation by the least common denominator (LCD) of all the fractions. Step 2Simplify. Simplify both sides of the equation by removing parentheses and other grouping symbols (if any) and combining like terms. © 2010 Pearson Education, Inc. All rights reserved 8

PROCEDURE FOR SOLVING LINEAR EQUATIONS IN ONE VARIABLE Step 3Isolate the Variable Term. Add appropriate expressions to both sides, so that when both sides are simplified, the terms containing the variable are on one side and all constant terms are on the other side. Step 4Combine Terms. Combine terms containing the variable to obtain one term that contains the variable as a factor. © 2010 Pearson Education, Inc. All rights reserved 9

PROCEDURE FOR SOLVING LINEAR EQUATIONS IN ONE VARIABLE Step 5Isolate the Variable. Divide both sides by the coefficient of the variable to obtain the solution. Step 6Check the Solution. Substitute the solution into the original equation. © 2010 Pearson Education, Inc. All rights reserved 10

EXAMPLE 3 Solving a Linear Equation Solve: Solution Step 2 Step 3 Step 4 © 2010 Pearson Education, Inc. All rights reserved 11

EXAMPLE 3 Solving a Linear Equation Solve: Solution continued The solution set is {3}. Step 5 Step 6 Check: © 2010 Pearson Education, Inc. All rights reserved 12

EXAMPLE 4 Solving a Linear Equation Solve: Solution Step 2 Step 3 © 2010 Pearson Education, Inc. All rights reserved 13

EXAMPLE 4 Solving a Linear Equation Solution continued 0 = 0 is equivalent to the original equation. The equation 0 = 0 is always true and its solution set is the set of real numbers. So the solution set to the original equation is the set of real numbers. The original equation is an identity. Solve: © 2010 Pearson Education, Inc. All rights reserved 14

TYPES OF LINEAR EQUATIONS There are three types of linear equations. 1.A linear equation that is satisfied by all values in the domain of the variable is an identity. For example, 2(x –1) = 2x – 2. 2.A linear equation that is not an identity, but is satisfied by at least one number is a conditional equation. For example, 2x = 6. © 2010 Pearson Education, Inc. All rights reserved 15

TYPES OF LINEAR EQUATIONS 3.A linear equation that is not satisfied for any value of the variable is an inconsistent equation. For example, x = x + 2 is an inconsistent equation. Since no number is 2 more than itself, the solution set of the equation x = x + 2 is . When you try to solve an inconsistent equation, you will obtain a false statement, such as 0 = 2. © 2010 Pearson Education, Inc. All rights reserved 16

RATIONAL EQUATIONS If at least one rational expression appears in an equation, then the equation is called a rational equation. © 2010 Pearson Education, Inc. All rights reserved 17

RATIONAL EQUATIONS © 2010 Pearson Education, Inc. All rights reserved 18 A solution that satisfies the new equation but does not satisfy the original equation is called an extraneous solution or extraneous root. So whenever we multiply an equation by an expression containing the variable, we must check all solutions obtained to reject extraneous solutions (if any).

EXAMPLE 5 Solving a Conditional Rational Equation Solve Solution Step 1 Step 2 © 2010 Pearson Education, Inc. All rights reserved 19

EXAMPLE 5 Solving a Conditional Rational Equation Solution continued Step 3 Step 4 © 2010 Pearson Education, Inc. All rights reserved 20 Step 5 The apparent solution is

EXAMPLE 5 Solving a Conditional Rational Equation Solution continued Step 6Check: © 2010 Pearson Education, Inc. All rights reserved 21 The solution set is

EXAMPLE 6 Solving a Rational Equation Solve for x: Solution Step 1 © 2010 Pearson Education, Inc. All rights reserved 22 Step 2

EXAMPLE 6 Solving a Rational Equation Solution continued © 2010 Pearson Education, Inc. All rights reserved 23 Step 4 Step 3 The equation 2 = 2 is always true for all values of x in its domain. The domain of x is all real numbers except −1 and 1. Therefore, the solution set is

EXAMPLE 7 Solving an Inconsistent Rational Equation Solve for m: Solution Step 1 © 2010 Pearson Education, Inc. All rights reserved 24 Step 2

EXAMPLE 6 Solution continued © 2010 Pearson Education, Inc. All rights reserved 25 Steps 3-5 Step 6Check: Because dividing by zero is undefined, we reject m = 3 as a solution. The solution set is . Solving an Inconsistent Rational Equation

EXAMPLE 8 Converting temperatures The formula for converting the temperature in degrees Celsius (C) to degrees Fahrenheit (F) is If the temperature shows 86° Fahrenheit, what is the temperature in degrees Celsius? Solution Substitute 86 for F. Solve for C. © 2010 Pearson Education, Inc. All rights reserved 26

EXAMPLE 8 Converting temperatures Solution continued Solve for C. Thus, 86º F converts to 30º C. © 2010 Pearson Education, Inc. All rights reserved 27

EXAMPLE 9 Solving for a Specified Variable Solvefor C. Solution © 2010 Pearson Education, Inc. All rights reserved 28

EXAMPLE 10 Bungee-Jumping TV Contestants Suppose that a television adventure program has its contestants bungee jump from a bridge 120 feet above the water. The bungee cord that is used has a % elongation, meaning that its extended length will be the original length plus a maximum of an additional 150% of its original length. The show’s producer wants to be sure that the jumper doesn’t get closer than 10 feet to the water, and no contestant will be more than 7 feet tall. How long can the bungee cord be? © 2010 Pearson Education, Inc. All rights reserved 29

EXAMPLE 6 Bungee-Jumping TV Contestants Solution Let x = length, in feet, of the cord to be used x + 1.5x = 2.5x = maximum extended length A cord length of 41.2 feet will meet all conditions. ++ Extended cord length Body length 10-foot buffer Height of bridge above the water = © 2010 Pearson Education, Inc. All rights reserved 30