Reasoning to Solve Equations and Inequalities


 Dominic Chambers
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1 Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing how operting cost nd ticket sle income were expected to relte to ticket price x. Cost: C(x) = 22, x Income: I(x) = 2,500x 50x 2 A grph of those two functions looks like this: 40,000 ConcertTicket Income nd Operting Cost 30,000 20,000 10, Concert Ticket Price 60 Think Aout This Sitution Questions importnt to plnning the concert cn e nswered y solving equtions nd inequlities involving the cost nd income functions. Wht would you lern from solutions of the following? I(x) = C(x) I(x) > C(x) I(x) < C(x) How would you solve the eqution nd inequlities in Prt using the following tools? Technology Resoning out the symolic function rules (without the help of clcultor tles or grphs) LESSON 4 REASONING TO SOLVE EQUATIONS AND INEQUALITIES 225
2 INVESTIGATION 1 Resoning out Liner Equtions nd Inequlities Situtions involving comprison of usiness plns often involve liner functions, leding to questions requiring the solution of liner equtions nd inequlities. For exmple, pizz compny considering lese options for delivery truck might hve choices like those shown elow. In ech cse, the lese cost (in dollrs) is function of lese time (in weeks). Pln A: A(t) = 2, t Pln B: B(t) = 1, t 1. Wht do the numers 2,500 nd 1,000 tell out the conditions of ech lese? Wht do the numers 75 nd 90 tell? 2. At the strt, Pln A is more expensive thn Pln B. To see when the lese costs might e equl, you could solve the eqution 2, t = 1, t. One wy to solve tht eqution is to reson like this: 2, t = 1, t 1, t = 90t 1,500 = 15t 100 = t. Justify ech step in the solution process.. How cn you check tht t = 100 is the solution? Wht does t = 100 tell out the trucklese plns? c. How could you rrive t the sme result with different sequence of steps? 3. Now, consider the inequlity 2, t < 1, t.. Wht will solution to this inequlity tell out the trucklesing sitution?. Why does ech step in the following resoning mke sense? 2, t < 1, t 2,500 < 1, t 1,500 < 15t 100 < t c. Wht does the solution 100 < t tell out the trucklesing sitution? d. How cn you check the solution? e. Use similr resoning to solve the inequlity 2, t > 1, t, nd explin wht the solution tells out the trucklesing sitution. 226 UNIT 3 SYMBOL SENSE AND ALGEBRAIC REASONING
3 In solving the equtions nd inequlities tht compre two trucklese plns, it is helpful to keep in mind wht the prts of ech expression men. Now try to use similr resoning ptterns to solve equtions nd inequlities without such clues. 4. Solve ech of the following equtions y symolic resoning lone. Record ech step in your resoning. Check your solutions.. 2x + 15 = 45 + x. 10 4x = 3x 4 c. 7x 11 = 10 d. 6x 15 = 4x + 10 e. 25 = 10 5x 5. Solve ech of the following inequlities y symolic resoning lone. Record ech step in your resoning. Check your nswers.. 3x + 10 < x > 23 c. 8x + 12 < 46 d x > Here re three solutions of inequlities tht led to incorrect results. In ech cse, show with function tles or grphs tht the proposed solutions re not correct. Then find nd correct the error in the resoning process.. Solve: 5x + 20 < 3x 20 < 2x 10 < x. Solve: 11x 19 < 15x x < 15x x < 36 x < 9 c. Solve: x < 3x x < 3x 18 < 6x 3 < x 7. Solve ech of the following inequlities y symolic resoning lone. Show ech step in your resoning. Check your nswers.. 3x + 10 < 5x x > 7x 13 c. 8x + 12 < 46 9x d x > 21x 15 LESSON 4 REASONING TO SOLVE EQUATIONS AND INEQUALITIES 227
4 Two equtions or inequlities re clled equivlent if they hve identicl solutions. One strtegy for solving liner equtions nd inequlities is to strt with the given eqution or inequlity nd construct sequence of simpler forms, ech equivlent to its predecessor, until you get n eqution or inequlity so simple tht the solution is ovious. The chllenge is to find wys of writing equivlent equtions nd inequlities tht do ecome progressively simpler. 8. Which of the following pirs of equtions nd inequlities re equivlent? Explin your resoning in ech cse.. 3x + 2 = 5 nd 3x = 3. 7x 8 = x nd 4x = 20 c. 1 x + 9 = 6 3 nd x + 9 = 18 d. 10x + 15 = 35 nd 2x + 3 = 7 e. 10x + 15 = 35 nd 10x = 20 f. 3x + 2 < 5 nd 3x < 3 g. 7x 8 > x nd 4x > 20 h. 10x + 15 < 35 nd 2x + 3 < 7 i. 10x + 15 > 35 nd 10x > Look ck over the pirs of equtions nd inequlities in Activity 8 nd your nswers to the equivlence question. Wht opertions on equtions nd inequlities seem likely to produce simpler equivlent forms? Checkpoint Mny situtions cll for compring two liner functions like the following: c f(x) = + x g(x) = c + dx Wht overll strtegy nd specific resoning steps would you use to solve n eqution of the form + x = c + dx? Explin how you could check the solution. Wht overll strtegy nd specific resoning steps would you use to solve n inequlity of the form + x < c + dx? How could you check the nswer? How do the grphs of expressions like y = + x nd y = c + dx illustrte solutions to the equtions nd inequlities descried in Prts nd? How would those solutions pper in tles of vlues for the two functions? Be prepred to explin your strtegies nd resoning to the entire clss. 228 UNIT 3 SYMBOL SENSE AND ALGEBRAIC REASONING
5 On Your Own Two cellulr telephone service plns offer monthly costs (in dollrs) tht re functions of time used (in minutes) with the following rules: B(t) = t C(t) = t Write nd solve (without use of technology) equtions nd inequlities tht help in nswering these questions:. Under wht conditions will Pln B cost less thn Pln C?. Under wht conditions will Pln B cost the sme s Pln C? c. Under wht conditions will Pln C cost less thn Pln B? In ech cse, show how you cn use clcultor to check your solutions. INVESTIGATION 2 Resoning out Qudrtic Equtions nd Inequlities Two key questions re often ssocited with qudrtic function models of the form f(x) = x 2 + x + c: Wht is the mximum (minimum) vlue nd where does it occur? For wht vlues of the input vrile x will f(x) = 0? In the cse of the concertplnning model descried t the strt of this lesson, the qudrtic income function ws I(x) = 2,500x 50x 2. The two key questions cn e stted in this wy: Wht ticket price will led to mximum projected income from ticket sles? Wht ticket prices will produce no projected ticket income t ll? You cn nswer oth questions y scnning tle or grph of (x, I(x)) vlues. But you cn lso get the nswers esily y using lgeric resoning. 1. Justify ech step in the following nlysis of the concertincome sitution.. Solving the eqution 2,500x 50x 2 = 0 will help.. The eqution in Prt is equivlent to 50x(50 x) = 0. c. 50x(50 x) = 0 when x = 0 or when x = 50. d. The mximum income will occur when x = 25. e. Tht income is $31,250. LESSON 4 REASONING TO SOLVE EQUATIONS AND INEQUALITIES 229
6 2. Unfortuntely, mny qudrtic equtions re not esy to solve using the type of fctoring tht worked so well in Activity 1. For exmple, consider the prolem of finding projected rekeven prices for the plnned concert. Those re the prices for which income from ticket sles will equl expenses for operting costs. Since the cost eqution for this sitution ws C(x) = 22, x, this prolem requires solving the eqution 2,500x 50x 2 = 22, x.. You cn strt y writing n equivlent eqution with qudrtic expression equl to 0: 50x 2 + 2,600x 22,500 = 0 Why is this eqution equivlent to the originl?. Fctor the left side of this eqution to get 50(x 2 52x + 450) = 0. Why is this fctored form equivlent to the eqution in Prt? The form of the eqution given in Prt does not look esy to continue to solve y fctoring! There is nother wy you cn solve qudrtic equtions, even when fctoring seems impossile. You cn use the qudrtic formul. If x 2 + x + c = 0 (nd 0), then the roots of the eqution re 4 c x = 2 or, writing these seprtely, + 4 c 2 2 ± c 2 2 x = 2 nd x = 2 (You ll explore derivtion of this formul in Lesson 5.) 3. Solve the rekeven eqution 50x 2 + 2,600x 22,500 = 0 using the qudrtic formul.. Give the vlues for,, nd c.. Evlute 2 c. Evlute 2 4 c 2 d. Now clculte x = c nd x = c 2 e. Descrie t lest three different wys to check your clculted roots in Prt d. Check the roots in the originl eqution using one of those methods. f. Use the qudrtic formul to solve the eqution x 2 52x = 0. Compre the result to the nswer in Prt d nd explin similrities or differences. 230 UNIT 3 SYMBOL SENSE AND ALGEBRAIC REASONING
7 4. Some computer softwre nd some clcultors hve solve feture tht llows you to solve equtions directly, if one side of the eqution is equl to 0. The procedure for using these solving cpilities vries. You my need to consult your mnul to lern how to use the feture. Use the solve feture on your clcultor or computer softwre to check your solutions to Prts d nd f of Activity Now consider the qudrtic eqution x 2 6x + 5 = 0. A grph of the function f(x) = x 2 6x + 5 is shown elow.. Give the vlues for,, nd c tht should e used to solve x 2 6x + 5 = 0 with the qudrtic formul.. Evlute 2 c. Evlute 2 4 c 2 4 c 4 c d. Now clculte x = 2 nd x = e. Compre the qudrtic formul clcultions to the grph of f(x) = x 2 6x + 5. Wht informtion is provided y the expression 2? By the expression 4 c? EQUATION SOLVER eqn : 0 = 50x x Use the qudrtic formul to solve ech of the following qudrtic equtions. Then try to solve the sme equtions y fctoring. In ech cse, check your work y using the solve feture on your clcultor or computer softwre or y sustituting your proposed roots for x into the eqution.. 2x 2 3x + 7 = 0. 5x 2 x 4 = 0 c. 3x 2 2x + 1 = 0 d. x 2 + 2x 3 = 0 e. 4x x + 9 = 0 LESSON 4 REASONING TO SOLVE EQUATIONS AND INEQUALITIES 231
8 7. Now look ck t your work in Activity 6 nd serch for connections etween the qudrtic formul clcultions nd the grphs of the corresponding function rules. Explin the specil significnce of the eqution x = 2 for qudrtic function with rule in the form f(x) = x 2 + x + c. Wht informtion is provided y the expression 2 4 c? 2 Test your ides in ech of the following cses y grphing the function nd the verticl line x = 2 (If you choose to use your clcultor rther thn sketching your grph, consult your mnul s needed.). f(x) = 2x 2 + 4x 9. f(x) = 3x 2 2x 5 c. f(x) = x 2 + 6x 10 d. f(x) = x 2 + 2x 9 8. Think out the wys in which the grph of f(x) = x 2 + x + c could intersect the xxis. How mny possile roots could the eqution x 2 + x + c = 0 hve?. Use the qudrtic formul to solve ech eqution nd identify the step tht first shows the numer of roots you cn expect. x 2 + 8x + 12 = 0 x 2 + 8x + 16 = 0 x 2 + 8x + 20 = 0. Sketch grphs of the qudrtic functions corresponding to the three equtions ove, nd explin how those grphs show the numer of roots in ech cse. 9. Suppose tht f(x), g(x), j(x), nd h(x) re qudrtic functions with the zeroes indicted elow. Find vlues of x for which ech of these functions would hve mximum or minimum vlues. Then write possile rules for the functions in fctored form.. f(6) = 0 nd f( 2) = 0. g( 7) = 0 nd g(3) = 0 c. j( 2) = 0 nd j( 5) = 0 d. h(2) = 0 nd h(4.5) = Explin how the qudrtic formul cn help you determine the minimum vlue of the function f(x) = 4x 2 7x UNIT 3 SYMBOL SENSE AND ALGEBRAIC REASONING
9 11. The qudrtic formul nd other equtionsolving methods cn e used to solve qudrtic inequlities s well. Exmine the grph of the function f(x) = 2x 2 5x 12 t the right. It hs zeroes t x = 1.5 nd x = 4.. Explin how the numer line grph elow shows the solution of the qudrtic inequlity 2x 2 5x 12 < How does the numer line grph in Prt relte to the grph of the function f(x) = 2x 2 5x 12? c. Mke numer line grph showing the solution of the qudrtic inequlity 2x 2 5x 12 > 0. d. How would you modify the numer line grph in Prt c to show the solution of the inequlity 2x 2 5x 12 0? Using Activity 11 s n exmple, if needed, solve the following qudrtic inequlities.. x 2 x 6 < 0. x 2 x 6 > 0 c. x 2 + 5x + 6 > 0 d. 2x 2 3x + 5 < 0 e. x 2 8x f. x 2 8x + 16 > 0 Checkpoint Qudrtic functions, with grphs tht re prols, cn e written with symolic rules in the form f(x) = x 2 + x + c. You hve lerned to solve the relted qudrtic equtions x 2 + x + c = 0 using the qudrtic formul. c d Explin the steps tht you would tke to determine the zeroes nd the minimum vlue of the function f(x) = 3x 2 2x 8. Wht re the dvntges nd disdvntges of solving qudrtic equtions y fctoring? By using the qudrtic formul? By using the solve feture of your clcultor or computer softwre? How cn you use the qudrtic formul or the solve feture to find fctored form of qudrtic function rule? How does use of the qudrtic formul show whether given eqution will hve 2, 1, or 0 roots? How will this informtion pper in grph? Be prepred to shre your methods nd thinking with the clss. LESSON 4 REASONING TO SOLVE EQUATIONS AND INEQUALITIES 233
10 On Your Own Use wht you hve lerned out the qudrtic formul to complete the following tsks.. Find the zeroes nd the lines of symmetry for the grphs of the following functions. f(x) = x 2 4x + 1 g(x) = x 2 + 6x 11 h(x) = x For ech of the following functions, find the minimum or mximum vlue of the function. f(x) = x 2 3x + 9 g(x) = x 2 + 8x + 2 h(x) = x 2 49 c. Grph solutions for these qudrtic inequlities: x 2 3x 4 > 0 x 2 + x 6 < 0 x 2 2x + 3 > UNIT 3 SYMBOL SENSE AND ALGEBRAIC REASONING
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