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1-1 - McGraw Hill Higher Education

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.

1-1 Linear Equations and Applications 7 Check 4 6 8 18 Sum of first three 4 6 8 8 Excess 10 Fourth Matched Problem 3Find three consecutive odd integers such that 3 times their sum is 5 …

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Transcription of 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.

2 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.

3 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.

4 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. 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.

5 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.

6 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. 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.

7 {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.]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.

8 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.

9 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.

10 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. 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.


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