1-1 - McGraw Hill Higher Education

1 21 Equations and InequalitiesOne of the important uses of algebra is the solving of equations andinequalities. In this chapter we look at techniques for solving linear andnonlinear equations and inequalities. In addition, we consider a numberof applications that can be solved using these techniques. Additional tech-niques for solving polynomial equations will be discussed in Chapter 1-1 Linear Equations and Applications Equations Solving Linear Equations A Strategy for Solving word problems Number and Geometric Problems Rate Time Problems Mixture Problems Some Final Observations on Linear EquationsAn algebraic equation is a mathematical statement that relates two algebraic expres-sions involving at least one variable. Some examples of equations with xas the vari-able areThe replacement set, or domain, for a variable is defined to be the set of numbersthat are permitted to replace the Domains of VariablesUnless stated to the contrary, we assume that the domain for a variable is theset of those real numbers for which the algebraic expressions involving the vari-able are real example, the domain for the variable xin the expression2x 4is R, the set of all real numbers, since 2x 4 represents a real number for all replace-ments of xby real numbers.

2 The domain of xin the equationis the set of all real numbers except 0 and 3. These values are excluded because theleft member is not defined for x 0 and the right member is not defined for x 2x 3 x 4 x 1 2x2 3x 5 0 11 x xx 2 3x 2 7 Equations1-1 Linear Equations and Applications3 The left and right members represent real numbers for all other replacements of xbyreal solution set for an equation is defined to be the set of elements in the domainof the variable that makes the equation true. Each element of the solution set is calleda solution,or root,of the equation. To solve an equationis to find the solution setfor the equation is called an identityif the equation is true for all elements from thedomain of the variable. An equation is called a conditional equationif it is true forcertain domain values and false for others. For example,2x 4 2(x 2)andare identities, since both equations are true for all elements from the respectivedomains of their variables.

3 On the other hand, the equations3x 2 5andare conditional equations, since, for example, neither equation is true for the domainvalue what we mean by the solution set of an equation is one thing; findingit is another. To this end we introduce the idea of equivalent equations. Two equa-tions are said to be equivalentif they both have the same solution set for a givenreplacement set. A basic technique for solving equations is to perform operations onequations that produce simpler equivalent equations, and to continue the process untilan equation is reached whose solution is of any of the properties of equality given in Theorem 1 will produceequivalent 1 Properties of EqualityFor a, b, and cany real a b, then a c b a b, then a c b a b, then ca cb, c a b, then , c a b, then either may replace the otherSubstitution Propertyin any statement without changing thetruth or falsity of the now turn our attention to methods of solving first-degree, or linear, equations inone variable.

4 Solving LinearEquationsac bc2x 1 1x5x2 3x 5x(x 3)41 Equations and InequalitiesDEFINITION 1 Linear Equation in One VariableAny equation that can be written in the formax b 0a 0 Standard Formwhere aand bare real constants and xis a variable, is called a linear, or first-degree, equation in one 1 2(x 3) is a linear equation, since it can be written in the standard form3x 7 1 Solving a Linear EquationSolve 5x 9 3x 7 and use the properties of equality to transform the given equation into an equivalentequation whose solution is equationAdd 9 to both like 3xfrom both like both sides by solution set for this last equation is obvious:Solution set: {8}And since the equation x 8 is equivalent to all the preceding equations in our solu-tion, {8} is also the solution set for all these equations, including the original equa-tion. [Note:If an equation has only one element in its solution set, we generally use the last equation (in this case, x 8) rather than set notation to represent thesolution.]

5 ]CheckOriginal equationSubstitute x each true statement 31 31 40 9 24 7 5(8) 9 3(8) 7 5x 9 3x 7 x 8 2x2 162 2x 16 5x 3x 3x 16 3x 5x 3x 16 5x 9 9 3x 7 9 5x 9 3x 71-1 Linear Equations and Applications5 Matched problem 1 Solve and check: 7x 10 4x 5We frequently encounter equations involving more than one variable. For exam-ple, if land ware the length and width of a rectangle, respectively, the area of therectangle is given by (see Fig. 1).A lwDepending on the situation, we may want to solve this equation for lor w. To solvefor w, we simply consider Aand lto be constants and wto be a variable. Then theequation A lwbecomes a linear equation in wwhich can be solved easily by divid-ing both sides by l:l 0 EXAMPLE 2 Solving an Equation with More Than One VariableSolve for Pin terms of the other variables: A P PrtSolutionThink of A, r, and tas to isolate both sides by 1 : 1 rt 0 Matched problem 2 Solve for Fin terms of C: C (F 32)A great many practical problems can be solved using algebraic techniques so many,in fact, that there is no one method of attack that will work for all.

6 However, we canformulate a strategy that will help you organize your for Solving Word the problem carefully several times if necessary that is, until youunderstand the problem , know what is to be found, and know what is given. A Strategy forSolving WordProblems59 P A1 rt A1 rt P A P(1 rt) A P Prtw AlA lwwlFIGURE 1 Area of a Equations and InequalitiesThe remaining examples in this section contain solutions to a variety of wordproblems illustrating both the process of setting up word problems and the techniquesused to solve the resulting equations. It is suggested that you cover up a solution, trysolving the problem yourself, and uncover just enough of a solution to get you goingagain in case you get stuck. After successfully completing an example, try the matchedproblem. After completing the section in this way, you will be ready to attempt afairly large variety of first examples introduce the process of setting up and solving word problems ina simple mathematical context.

7 Following these, the examples are of a more sub-stantive 3 Setting Up and Solving a Word ProblemFind four consecutive even integers such that the sum of the first three exceeds thefourth by x the first even integer, thenxx 2x 4andx 6represent four consecutive even integers starting with the even integer x. (Remem-ber, even integers increase by 2.) The phrase the sum of the first three exceeds thefourth by 8 translates into an equation:The four consecutive integers are 4, 6, 8, and 10. x 4 2x 8 3x 6 x 14 x (x 2) (x 4) (x 6) 8 Sum of the first three Fourth Excess Number andGeometric one of the unknown quantities be represented by a variable, say x, andtry to represent all other unknown quantities in terms of x. This is an impor-tant step and must be done appropriate, draw figures or diagrams and label known and unknown for formulas connecting the known quantities to the an equation relating the unknown quantities to the known the equation and write answers to allquestions asked in the and interpret all solutions in terms of the original problem not justthe equation found in step 5 since a mistake may have been made in set-ting up the equation in step Linear Equations and Applications7 Check4 6 8 18 Sum of first three4 6 8 8 Excess10 FourthMatched problem 3 Find three consecutive odd integers such that 3 times their sum is 5 more than 8 timesthe middle 1 According to property 1 of Theorem 1, multiplying both sides of an equation bya nonzero number always produces an equivalent equation.

8 By what number wouldyou choose to multiply both sides of the following equation to eliminate all thefractions?If you did not choose 12, the LCD of all the fractions in this equation, you couldstill solve the resulting equation, but with more effort. (For a discussion of LCDsand how to find them, see Section A-4.)EXAMPLE 4 Using a Diagram in the Solution of a Word ProblemIf one side of a triangle is one-third the perimeter, the second side is one-fifth theperimeter, and the third side is 7 meters, what is the perimeter of the triangle?SolutionLet p the perimeter. Draw a triangle and label the sides, as shown in Figure 2. ThenMultiply both sides by 15, the LCD. This and the next step usually can be done perimeter is 15 meters. p 15 7p 105 15p 5p 3p 105 15p 15 p3 15 p5 15 7 15 p 15 p3 p5 7 p p3 p5 7 p a b cx 13 x4 12acbp a b c7 metersp3p5 FIGURE 281 Equations and InequalitiesCheckMatched problem 4If one side of a triangle is one-fourth the perimeter, the second side is 7 centimeters,and the third side is two-fifths the perimeter, what is the perimeter?

9 CAUTIONA very common error occurs about now students tend to confuse algebraicexpressionsinvolving fractions with algebraic equationsinvolving these two problems:(A) Solve: (B) Add: The problems look very much alike but are actually very different. To solvethe equation in (A) we multiply both sides by 6 (the LCD) to clear the frac-tions. This works so well for equations that students want to do the samething for problems like (B). The only catch is that (B) is not an equation, andthe multiplication property of equality does not apply. If we multiply (B) by6, we simply obtain an expression 6 times as large as the original! Comparethe following:(A)There are many types of quantity rate time problems and distance rate time prob-lems. In general, if Qis the quantity of something produced (kilometers, words , parts,and so on) in Tunits of time (hours, years, minutes, seconds, and so on), then theformulas given in the box are relevant.

10 Rate TimeProblems x 12 5x 60 3x 2x 60 6 x2 6 x3 6 10 x2 x3 10x2 x3 10x2 x3 10153p3 5155p5 3715 metersSide 1 Side 2 Side 3 Perimeter(B) 5x 606 3x6 2x6 606 3 x3 2 2 x2 3 6 106 1x2 x3 101-1 Linear Equations and Applications9 Quantity Rate Time FormulasIf Qis distance D, then[Note: Ris an average or uniform rate.]EXAMPLE 5A Distance Rate Time ProblemThe distance along a shipping route between San Francisco and Honolulu is 2,100nautical miles. If one ship leaves San Francisco at the same time another leaves Hon-olulu, and if the former travels at 15 knots* and the latter at 20 knots, how long willit take the two ships to rendezvous? How far will they be from Honolulu and SanFrancisco at that time?SolutionLet T number of hours until both ships meet. Draw a diagram and label knownand unknown parts. Both ships will have traveled the same amount of time when theymeet.