2 And Problem Solving Equations, Inequalities,

CHAPTER2Equations, Inequalities,and Problem Solving2.1 Linear Equations in OneVariable2.2 An Introduction toProblem Solving2.3 Formulas and ProblemSolving2.4 Linear Inequalities andProblem SolvingIntegrated Review—Linear Equations andInequalitiesToday, it seems that most people in the world want to stay connected most of the time.In fact, 86% of U.S. citizens own cell phones. Also, computers with Internet access arejust as important in our lives. Thus, the merging of these two into Wi-Fi-enabled cellphones might be the next big technological explosion. In Section 2.1, Objective 1, andSection 2.2, Exercises 35 and 36, you will find the projected increase in the number ofWi-Fi-enabled cell phones in the United States as well as the percent increase. (Source:Techcrunchies.com)Number of Wi-Fi-Enabled Cell Phonesin the U.S. (in millions)Projected Growth of Wi-Fi-Enabled Cell Phones in the 12012201320142.5 Compound Inequalities2.6 Absolute ValueEquations2.7 Absolute ValueInequalitiesMathematics is a tool for solvingproblems in such diverse fieldsas transportation, engineering,economics, medicine, business, andbiology. We solve problems usingmathematics by modeling real-worldphenomena with mathematicalequations or inequalities. Our abilityto solve problems using mathematics,then, depends in part on our ability tosolve equations and inequalities. Inthis chapter, we solve linear equationsand inequalities in one variable andgraph their solutions on number lines.2015Year47

48 CHAPTER 22.1Equations, Inequalities, and Problem SolvingLinear Equations in One VariableOBJECTIVEOBJECTIVES1 Solve Linear Equations UsingProperties of Equality.2 Solve Linear EquationsThat Can Be Simplified byCombining Like Terms.3 Solve Linear Equations1Solving Linear Equations Using Properties of EqualityLinear equations model many real-life problems. For example, we can use a linearequation to calculate the increase in the number (in millions) of Wi-Fi-enabled cellphones.Wi-Fi-enabled cell phones let you carry your Internet access with you. There arealready several of these smart phones available, and this technology will continue toexpand. Predicted numbers of Wi-Fi-enabled cell phones in the United States for various years are shown below.Containing Fractions orDecimals.149150138140130118120110(in millions)Equations with No Solution.Number of Wi-Fi-Enabled Cell Phones in the U.S.4 Recognize Identities andProjected Growth of Wi-Fi-Enabled Cell Phones in the 01320142015YearTo find the projected increase in the number of Wi-Fi-enabled cell phones in the UnitedStates from 2014 to 2015, for example, we can use the equation below.In words:Increase incell phonesiscell phones in2015minuscell phones in2014Translate:x 149-138Since our variable x (increase in Wi-Fi-enabled cell phones) is by itself on one side ofthe equation, we can find the value of x by simplifying the right side.x 11The projected increase in the number of Wi-Fi-enabled cell phones from 2014 to 2015is 11 million.The equation x 149 - 138, like every other equation, is a statement that twoexpressions are equal. Oftentimes, the unknown variable is not by itself on one sideof the equation. In these cases, we will use properties of equality to write equivalentequations so that a solution may be found. This is called solving the equation. In thissection, we concentrate on solving equations such as this one, called linear equationsin one variable. Linear equations are also called first-degree equations since the exponent on the variable is 1.Linear Equations in One Variable3x -157 - y 3y4n - 9n 6 0z -2

Section 2.1Linear Equations in One Variable 49Linear Equations in One VariableA linear equation in one variable is an equation that can be written in the formax b cwhere a, b, and c are real numbers and a 0.When a variable in an equation is replaced by a number and the resulting equationis true, then that number is called a solution of the equation. For example, 1 is a solution of the equation 3x 4 7, since 3112 4 7 is a true statement. But 2 is not asolution of this equation, since 3122 4 7 is not a true statement. The solution setof an equation is the set of solutions of the equation. For example, the solution set of3x 4 7 is 5 1 6 .To solve an equation is to find the solution set of an equation. Equations with thesame solution set are called equivalent equations. For example,3x 4 73x 3x 1are equivalent equations because they all have the same solution set, namely 5 1 6 . Tosolve an equation in x, we start with the given equation and write a series of simplerequivalent equations until we obtain an equation of the formx ⴝ numberTwo important properties are used to write equivalent equations.The Addition and Multiplication Properties of EqualityIf a, b, and c, are real numbers, thena b and a c b c are equivalent equations.Also, a b andac bc are equivalent equations as long as c 0.The addition property of equality guarantees that the same number may be added toboth sides of an equation, and the result is an equivalent equation. The multiplicationproperty of equality guarantees that both sides of an equation may be multiplied bythe same nonzero number, and the result is an equivalent equation. Because we definesubtraction in terms of addition 1a - b a 1 -b2 2, and division in terms of multia1plication a a # b , these properties also guarantee that we may subtract the samebbnumber from both sides of an equation, or divide both sides of an equation by thesame nonzero number and the result is an equivalent equation.For example, to solve 2x 5 9, use the addition and multiplication propertiesof equality to isolate x—that is, to write an equivalent equation of the formx ⴝ numberWe will do this in the next example.EXAMPLE 1Solve for x: 2x 5 9.Solution First, use the addition property of equality and subtract 5 from both sides.We do this so that our only variable term, 2x, is by itself on one side of the equation.2x 5 92x 5 - 5 9 - 5 Subtract 5 from both sides.2x 4Simplify.Now that the variable term is isolated, we can finish solving for x by using the multiplication property of equality and dividing both sides by 2.2x4 22x 2Divide both sides by 2.Simplify.

50 CHAPTER 2Equations, Inequalities, and Problem SolvingCheck: To see that 2 is the solution, replace x in the original equation with 2.2x 5 2122 5 ⱨ4 5ⱨ9 9 Original equation9 Let x 2.99 TrueSince we arrive at a true statement, 2 is the solution or the solution set is 5 2 6 .PRACTICE1Solve for x: 3x 7 22.EXAMPLE 2Solve: 0.6 2 - 3.5c.Solution We use both the addition property and the multiplication property of equality.0.60.6 - 2-1.4-1.4-3.50.4Helpful HintDon’t forget that0.4 c and c 0.4 areequivalent equations.We may solve an equation so thatthe variable is alone on either sideof the equation. 2 - 3.5c 2 - 3.5c - 2 Subtract 2 from both sides. -3.5cSimplify. The variable term is now isolated.-3.5c Divide both sides by -3.5.-3.5- 1.4 cSimplify.-3.5Check:0.60.60.60.6 2 - 3.5cⱨ 2 - 3.510.42ⱨ 2 - 1.4 0.6Replace c with 0.4.Multiply.TrueThe solution is 0.4, or the solution set is 5 0.4 6 .PRACTICE2Solve: 2.5 3 - 2.5t.OBJECTIVE2Solving Linear Equations That Can Be Simplified by Combining LikeTermsOften, an equation can be simplified by removing any grouping symbols and combining any like terms.EXAMPLE 3Solve: -4x - 1 5x 9x 3 - 7x.Solution First we simplify both sides of this equation by combining like terms. Then,let’s get variable terms on the same side of the equation by using the addition propertyof equality to subtract 2x from both sides. Next, we use this same property to add 1 toboth sides of the equation.-4x - 1 5xx - 1x - 1 - 2x-x - 1-x - 1 1-x 9x 3 - 7x2x 3Combine like terms.2x 3 - 2x Subtract 2x from both sides.3Simplify.3 1Add 1 to both sides.4Simplify.Notice that this equation is not solved for x since we have -x or -1x, not x. To solvefor x, we divide both sides by -1.-x4 -1-1x -4Divide both sides by -1.Simplify.

Section 2.1Linear Equations in One Variable 51Check to see that the solution is -4, or the solution set is 5 -4 6 .PRACTICESolve: -8x - 4 6x 5x 11 - 4x.3If an equation contains parentheses, use the distributive property to remove them.EXAMPLE 4Solve: 21x - 32 5x - 9.Solution First, use the distributive property.2(x-3) 5x-92x - 6 5x - 9Use the distributive property.Next, get variable terms on the same side of the equation by subtracting 5x from bothsides.2x - 6 - 5x-3x - 6-3x - 6 6-3x-3x-3x 5x - 9 - 5x Subtract 5x from both sides.-9Simplify.-9 6Add 6 to both sides.-3Simplify.-3 Divide both sides by - 3.-3 1Let x 1 in the original equation to see that 1 is the solution.PRACTICE4Solve: 31x - 52 6x - 3.OBJECTIVE3Solving Linear Equations Containing Fractions or DecimalsIf an equation contains fractions, we first clear the equation of fractions by multiplyingboth sides of the equation by the least common denominator (LCD) of all fractions inthe equation.EXAMPLE 5Solve for y:yy1- .346Solution First, clear the equation of fractions by multiplying both sides of the equation by 12, the LCD of denominators 3, 4, and 6.yy1- 346yy1- b 12a b Multiply both sides by the LCD 12.346yy12a b - 12a b 2Apply the distributive property.34Simplify.4y - 3y 212 ay 2Simplify.Check: To check, let y 2 in the original equation.yy1- 346221- ⱨ346Original equation.Let y 2.

52 CHAPTER 2Equations, Inequalities, and Problem Solving86 ⱨ1121262 ⱨ112611 66Write fractions with the LCD.Subtract.Simplify.This is a true statement, so the solution is 2.PRACTICE5Solve for y:yy1- .254As a general guideline, the following steps may be used to solve a linear equationin one variable.Solving a Linear Equation in One VariableStep 1.Step 2.Step 3.Step 4.Step 5.Step 6.Clear the equation of fractions by multiplying both sides of the equation bythe least common denominator (LCD) of all denominators in the equation.Use the distributive property to remove grouping symbols such asparentheses.Combine like terms on each side of the equation.Use the addition property of equality to rewrite the equation as an equivalentequation with variable terms on one side and numbers on the other side.Use the multiplication property of equality to isolate the variable.Check the proposed solution in the original equation.EXAMPLE 6Solve for x :x 51x - 3 2x .228Solution Multiply both sides of the equation by 8, the LCD of 2 and 8.x 51 b22x 518ab 8#2241x 52 44x 20 44x 24-11x 24-11x-11x-118aHelpful HintWhen we multiply both sides ofan equation by a number, thedistributive property tells us thateach term of the equation is multiplied by the number.x - 3b8x-38 # 2x - 8ab816x - 1x - 3216x - x 315x 33-21-21-112111 8a 2x -Multiply both sides by 8. Apply the distributive property. x Simplify.Use the distributive property to remove parentheses.Combine like terms.Subtract 15x from both sides.Subtract 24 from both sides.Divide both sides by -11.Simplify.21Check: To check, verify that replacing x withmakes the original equation true.1121The solution is .11PRACTICE6Solve for x: x -x - 2x 31 .1244