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# Class Note for MATH 1310 at UH

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Date Created: 02/06/15
Secl39nn 27 Sullving Nuruinear lmquzlil39zs Page I um Solving Nonlinear Inequalities A nonllnear lnequallty ean olten be solved by a number line analysis In thrs methool the real number llne ls dlvlded rnto rntervals Ifthe rnequalrty lnvolves a prooluet then the slgns ofrts factors are eheekeolto oletermrne the slgn ofthatproduct m eaeh ofthe rntervals Ifthe rnequalrty lnvolves a quotrent thenthe slgns oflts numerator anol olenomrnator are eheekedto oletermrne the slgn ofthat quotrent m eaeh of the rntervals Thls teehnrque for solvmg a nonlmear rnequalrty ls lllustxated m the example problems that follow Example 1 Solve the rnequahty x2 e x gt o lee the answer m rnterval notauon and graph the soluuon set Snlutinn tn Example 1 Set one slde ofthe rnequalrty equal to 0 ens Exes gtoeo suhtmetonum hothauestosettheRlls enuutou x exeo gt0 Slmphfy r2re3gto Freturtheuls z Solve the eorresponolrng equatron r2re3 0 r2re3o Apply the Zeruprouuct Property 3 Place these numbers on a number 11113 Secl39nn 27 Snlving Nunlinear lmquzlil39zs Pig 2 quot22 The open melee at e 2 and 3 mdmate that these numbers do not satisfy the inequality x 2 3 gt 0 72 2x727 3 0X75 00 3233 lt5gtltogt00 4 Let the intervals determmed by 72 and 3 723 loo l X e e 72 3 5 Checkthe Sign oftheproduct x2xe3 onthese intervals The 1nterval7w72 Ifxxs m the interval Ha z then x lt72 We eah choose atest heme sueh as x 7310 check the Sign on the factors rd 2 and x7 3 Letx shgh ofx2 shgh ofxe3 gh owe 2x3 The interval 723 Ifxxs m the interval 723 then 72 x lt3 We eah choose atest heme sueh as x 0 to checkthe Sign on the factors x2 andT7 3 Let x2 x73 2gt0 3lt0 shgh ofx 2 shgh one 3 e S EHiELWLZJ Secl39nn 27 Snlving Nunlinear lmquzlil39zs Pig 3 M22 The interval 3w Ifxxs m the interval 3w then x gt 3 We can choose atest value sueh as x 4 to checkthe srgh on the factors x2 andT7 3 Letx 4 x2426gt0 x734731gt0 Srgh ofx2 Srgh ofre3 rgh o re2r3 T 5 Determine the soluuor ofthe rhequamy r2re3 gt 0 by usmg the number hhe analysts The soluuor rh interval notation rs 7w72v3w The gaph ofthe soluuor rs sketched below e e 72 3 Example 2 Solve the rhequamy so Grve the answenn interval notation andgraph the soluuoh set Snllltinn m Example 1 Fmdthe values ofx for whreh the numerator rs 0 0 x711eo1 Audmhmhsrues x shuphry x71 Secl39nn 27 Sulving Nunlinear lmquzlil39zs Pig e quot22 z Fmdthe values ofx for whxch the denommator 15 0 xe2eo xe22eo2 Addltabmhsxdes x2 snpury 3 Place these numbers on a number hne e 1 2 The closed ene1e at 1 1nd1cates that 1 saus es the mequahty P 0 X g 0 The open ene1e at 21nd15ates that 2 does not sausfythe x71 mequahty Notethatsubsututmg 21ntothe expressxon would put 0 In the denommator 4 Lst the mtervals determmed by 1 and 2 7w 1 1 2 200 o e 1 2 5 Cheek the sxgn ofthe quouent XL on each ofthese mtervals The mterval em Ifsz 1n the mterval 7001 then x lt1 We can choose atest value sueh as x 0 to checkthe sxgn of the numeracmq andthe denommamhxe 2 Lax0 x71 0 e 1 lt 0 x7207272lt0 5min 27 Snlving Nunlinear lmquzlil39ls P2ge uf22 5 1gnofx 1 ngqofx72 7 e 2 ngnof X 1 1 The mterval 1 2 Ifsz m the mterval 12111121110 lt2 We can choose atest value such 3 as x E to check the sxgn ofthe numeratortil andthe denommatorxi 2 51g ofxil ngqofx72 7 e 7 1 2 ngnof X 1 7 X The mterval 201 Ifsz m the mterval 201 then x 2 We can choose atest value such as x 3 to check the sxgn othe numerator 17 1 andthe denommatorxi 2 Letx x7173712gt0 x723721gt0 51g ofxil 519 on 2 O e x71 1 2 519 of Secl39nn 27 Snlving Nunlinear lmquzlil39zs Pig 6 quot22 g 0 by usmg the number hhe 5 Deterrmhe the solutmh ofthe thequahty analysts e o e The solutror rh rhterval notatror rs 12 The gaph ofthe solutror rs sketehed below 9 1 2 Example 3 Solve the thequahty x4x7 3 lt 0 whte the solutmh rh rhterval notation and graph the solutmh on therea1 humberhhe Snllltinn tn Example Solve the conespondmg equahor forx r4rezo x40 or reko x 4 x3 Place 74 and 3 on ahumberhhe The open erreles mdmate that the numbers do not satrsfy the thequahty x4x7 3 lt0 Secl39nn 27 Snlving Nunlinear lmquzlil39ls Pig 7 quot22 List the mtemls detexmmed by 74 and 3 74 300 Cheekthe stghs ofthe factors x4 and73 on the mteml 7 74 by usmg atestvalue 75 is m the mteml 7094 For x X4 5471lt0 and x73757378 lt0 Detemhhe the stgh ofthe pmduet x4xe3 on the mteml 4943 stgh ofx4 stgh one stgh ofltx4xe3 Show the results on the number hhe e e Check the stghs ofthe factors x4 and73 on the mteml 43 by usmg atestvalue Ulsm the mteml 743 Forx x4044gt0 and x7307373lt0 Secl39nn 27 Snlving Nunlinear lmquzlil39zs Pig 3 quot22 Deterrmhe the ugh ofthe produet r4re3 on the rhterval 743 Srgh ofx4 Srgh one 7 Srgh of r4re3 7 Show the results on the number hhe e e 74 3 Check the srghs of the factors x4 mare 3 on the rhterval ma by usmg atestvalue 4 rs u the rhterval 300 Fmr x4448gt0 md x734731gt0 Deterrmhe the ugh ofthe produet r4re3 on the rhterval 300 Srgh ofx4 Srgh one Srgh of r4re3 Show the results on the number hhe whte the soluuor to the rhequahty x4x7 3 lt0 from the number hhe analysis Secl39nn 27 Snlving Nunlinear lmquzlilis Page 9 quot22 The solution m mterval notation rs 43 The graph ofthe soluttoh rs sketched below e e 74 3 Example 4 Solve the mequahty x2 4x2 32 whte the solution m rhterva1 hotatror and graph the sotutron on the rea1 number hhe Snllltinn tn Example 4 To solve the mequahty for r begrh by settmg one state equal to 0 x2 4x232 x2 4re322 32732 x2 4xr 32 2 0 Factor the LHS x x74 2 0 Solve the comespondmg equataor forx x8x74 0 x8 0 or re r 8 Place 78 and4 on ahumberhhe o o 78 4 The closed etreles mdmate that the numbers sausfy the mequahty ramp 4 2 0 Secl39nn 27 Sullilg Nunlinearlnzquzlilis Pig I uf22 Ltst the mtem1s detehmmed by e 8 and 4 701178 400 Cheekthe stghs ofthe factors x8 and74 on the mteml ew x by usmg atestvalue 79 is m the thteml 70078 Forx x879871lt0 and x747974713 lt0 Detemhhe the stgh ofthe pmduet x8x74 on the mteml 4078 stgh ofx8 e stgh ofx74 e stgh of x8x74 Show the results on the number hhe o o 78 4 Cheekthe stghs ofthe factors x8 and74 on the mteml 784 byusmg atestvalue 0 is m the thteml 784 Forx0y x8088gt0 and x7407474lt0 Secl39nn 27 Sullilg Nunlinearlnzquzlilis Fig 11 M22 Detemuhe the srgh ofthe produet x 8x 74 on the rhterval 784 Srgh ofx8 Srgh ofxrzt e Srgh ourmmzz 7 Show the results on the number hhe 78 4 Check the srghs ofthe factors x8 shards on the rhterval 4w by usmg atestvalue 5 is m the mterval Atom Forx5 x85813gt0 and x7457471gt0 Detemuhe the srgh ofthe produet x8x74 on the rhterval 400 Srgh ofx8 Srgh ofxrzt Srgh of x8x74 Show the results on the number hhe whte the soluuor to the rhequahty x8x74 20 from the number hhe analysts Secl39nn 27 Sullilg Nunlinearlnzquzlilis Pzg212uf22 The soluhon m rnterya1 notahon rs 7w78v4w The graph ofthe solutron rs sketehed below 4 O O gt 4 Example 5 2 Solve the rnequahty X gt 0 wrrte the soluhon m rnterya1 notahon x and graph the solutron on the rea1 nurnberhne Snlntinn tn Example 5 To solve the rnequahty forx begm by fmdmg the values ofx where both the numerator and denornrnator are 0 To nd where the nurneratorrs 0 solve the equanonx2 0 by subtraehng 2 from both srdes rd 2 e 2 0 e 2 To nd where the denornrnatorrs 0 solve the equahon x75 0 by addmg 5 to both srdes x75505 x 5 Place 72 and 5 on anurnberhne e e 2 5 The open erreles rndeeate that the numbers do not satrsfy the rnequahty x2 x75 gt0 Secl39nn 27 Sullilg Nunlinearlnzquzlilis Pig mtzz Let the mtervals detexmmed by 7 2 and 5 7w 2 5 5amp0 l 72 5 cheekthe Signs ofthe numeratorx2 and the denommatom75 on the mteryal 7w72 by usmg atestvalue 735quot the mteryal 7w72 Forx 73 x2 3271lt0 375778 lt0 x2 Detemhhe the stgh ofthe quoheht on the mteryal 7w72 x75 stgh ofx2 7 stgh ofx75 7 x2 stgh of x75 Show the results on the number lme e e 72 5 cheekthe Signs ofthe numeratorx2 and the denommatom75 on the mteryal 725 by usmg atestyame 0 is m the mteryal 725 Forx0y x2022gt0 and x757075775lt0 Secl39nn 27 Sullilg Nunlinearlnzquzlilis Pig Nuf22 x2 Deterrmhe the srgh ofthe quoheht on the rhterval 725 X Srgh ofx2 Srgh ofres e Srgh of 2 X Show the results on the number lme e e 2 5 Check the srghs of the numerator x2 and the denommator x75 on the mterval 500 by usmg atest value 6 rsrh the rhterval 500 Forx6 x2628gt0 and x756751gt0 x2 Deterrmhe the srgh ofthe quoheht on the rhterval 500 X Srgh ofx2 Srgh ofres x2 Srgh of x Show the results on the number hhe Seernn 27 Sullilg Nunlinearlnzquzlilis Page ISuf22 x2 Wnte the soluhon to the rnequahty gt 0 from the number hne analysts X The soluhon m rnterya1 notataon rs 7w72u5w The graph ofthe solutron rs sketehed below Example Solve the rnequahty X 71 7 so wrrte the soluuon m tnteryal notation X and graph the solutron on the rea1 number hne Snllltinn tn Example To solve the rnequahty for x begm by fmdmg the values ofx where both the numerator and denornrnator are 0 To nd where the nurneratorrs 0 solve the equataon x71 0 by addmg 1 to both states rem 01 x1 To nd where the denornrnator rs 0 solve the equataon x77 0 by addmg 7 to both states re 77 07 x7 Place 1and7 on a numberlme o e Secl39nn 27 Suiting Nunlinearlnzquzlilis Pig I uf22 x71 The closed errele at 1 mdmates that rt satrsfres the rnequahty 0 The X open errele at7 rndreates thatrt does not sausfy the rnequahty Lrst the rntervals deterrmned by 1 and 7 00 1 1 7 700 Check the srgns ofthe numerator x71 andthe denomrnatorxJ on the rnterval em by usmg atest value 0 rs m the rnterval em Forx 0 x7107 la and x77 e7e7lt0 Deterrnrne the srgn ofthe quotient 71 on the rnterva1 em X Srgnofrel e SrgnofrJ e SrnofX g x77 Show the results on the number hne Check the srgns ofthe numerator x71 andthe denomrnatorxJ on the rnterval 17 by usmg atestvalue Secl39nn 27 Sullilg Nunlinearlnzquzlilis Pig mnz 2lsmthelnterva1 17 Forx2 x712711gt0 and x7727775lt0 x71 Detemhhe the stgh ofthe quoheht on the thteml 17 X stgh ofxel stgh ofxe7 e stgh of Show the results on the number hhe Check the stghs of the numeratomel andthe denommatom on the mteml 7w by usthg atestvehte 8xsmthemterva17w Forx8 x718717gt0 and x778771gt0 on the thteml 7w Detemhhe the stgh ofthe quoheht stgh ofxel stgh ofxe7 Secl39nn 27 stating Nunlinearhlzquzlil39ls Pig IEufZZ Show the results on the number 1the x71 whte the solution to the thequahty 0 tom the humbet 1the analysxs X The solution th thtetva1 notation ts 17 The graph othe sohttton ts sketehed below 9 1 7 Example 7 1 1 Solve the mequahty e lti1 whte the so1uttonththtem1 notation x x7 and graph the sohttton on the tea1 humbethhe Snllltinn tn Example 7 To solve the mequahty at x begth by settmg one stde equal to 0 1 E l x Secl39nn 27 Sullilg Nunlinearlnzquzlilis Pig warzz Fmdthe values ofx where the dehommatons 0 x071 0 x 0 0h x71 0 F1 Place 0 and 1 on anumber hhe e e 0 1 The open melee mdmate that these numbers do not sausz the mequahty List the mtemls detemhhed by 0 and 1 7w 0 L0 1 W Check the stgh othe denominator x971 on the mteml emu by usmg atestvalue 71 is m the interval emu For x0717 77172 0 Detemuhe the stgh ofthe quotient on the interval emu x971 stgh mm 1 stgh of x071 Secl39nn 27 Sullilg Nunlinear lmquzlil39ls Show the results on the number hhe e e 0 1 Check the Sign ofthe denommator xx71 on the mterval 01 by usmg atestvalue 15m the mterval 01 Forx 1 1 1 5J Determme the Sign ofthe quotient on the mterval 01 xx71 stgh mm 1 e1 xx71 stgh of Show the results on the number lme e e 0 1 Check the Sign ofthe denommatmmn on the mterval 1w by usmg atestvalue 2 is m the mterval Lou Forx P2ge2 uf22 Secl39nn 27 stating Nunlinearhlzquzlil39ls Page 21 M22 Detenntne the stgn ofthe quottent on the tntemal Lou m4 stgn own 1 stgn of m4 Show the results on the number 1tne Wnte the sohttaon to the tneqttahty lt 0 torn the numberlme analysxs men The sohttaon tn tntemal notataon ts mmLou The graph ofthe sohttton ts sketehed below n e h 0 1 Exercise Set 27 Nonlinear Inequalities Solve each of the following inequalities graph the solution set on the real number line and then write the solution in interval notation 1 10 11 12 13 14 15 16 17 18 19 20 21 22 x3x750 x72x7720 4x3lix lt 0 57x3xilgt0 x277x10gt0 x23x770 0 x2 2 4x21 x2 lt 4x3 8x275x lt 2x24 4x25gtx216x x2 gt25 x2 le 2 725 x278x 716 x72x74xl20 x37xx50 x4xil2 gt0 x7 2x32 g 0 24 25 26 27 28 29 30 31 32 33 34 35 36 23 2xllt0 x4 x76 gt0 273x 5x72 g 4 x3 47x gt 73 xl x 4 lt x75 3 x2 2 2 x77 5 3 5 2 x74 x6 2 3 7 g x71 x4 312 7 2 x x2 if 2 273 x71 x4 x75 x2 xl x73 x47x3 gt 0 x72 x75 x4lt 9x2 x5 2 x3 Answer the following 37 38 A rectangle is 5 inches longer than it is Wide Find all possible measurements for the Width of the rectangle if its area is greater than 84 square inches Paul has 300 feet of fence to enclose a rectangular plot of land for a garden If Paul wants the garden to have an area of at least 5400 square feet What is the range of values possible for the Width of the garden

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