3.6: Tatua Maombi na Usawa wa Mstari

Malengo ya kujifunza

Mwishoni mwa sehemu hii, utaweza:

• Tatua programu na kutofautiana kwa mstari

Kumbuka

Kabla ya kuanza, fanya jaribio hili la utayari.

1. Andika kama usawa: x ni angalau 30.
   Kama amekosa tatizo hili, kupitia Zoezi 2.7.34.
2. Kutatua\(8−3y<41\).
   Kama amekosa tatizo hili, kupitia Zoezi 2.7.22.

Tatua Maombi na Usawa wa Mstari

Hali nyingi za maisha halisi zinahitaji sisi kutatua kutofautiana. Kwa kweli, maombi ya kukosekana kwa usawa ni ya kawaida kwamba sisi mara nyingi hata kutambua sisi ni kufanya algebra. Kwa mfano, ngapi galoni za gesi zinaweza kuwekwa kwenye gari kwa $20? Je, kodi ya ghorofa ya bei nafuu? Je, kuna muda wa kutosha kabla ya darasa kwenda kupata chakula cha mchana, kula, na kurudi? Ni kiasi gani cha fedha kinachopaswa gharama ya zawadi ya kila mwanachama wa familia bila kwenda juu ya bajeti?

Njia tutakayotumia kutatua programu na kutofautiana kwa mstari ni kama ile tuliyotumia wakati tulitatua programu na equations. Tutasoma tatizo na kuhakikisha maneno yote yanaeleweka. Ifuatayo, tutatambua kile tunachotafuta na kugawa variable ili kuiwakilisha. Tutaelezea tena tatizo katika sentensi moja ili iwe rahisi kutafsiri kwa usawa. Kisha, sisi kutatua kukosekana kwa usawa.

Zoezi\(\PageIndex{1}\)

Emma got kazi mpya na itakuwa na hoja. Mapato yake ya kila mwezi yatakuwa $5,265. Ili kuhitimu kukodisha ghorofa, mapato ya kila mwezi ya Emma lazima iwe angalau mara tatu kama kodi. ni kodi ya juu Emma kuhitimu nini?

Jibu

\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the highest rent Emma will qualify for}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let r = rent}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate} \text{ into an inequality.}} &{} \\{} &{\text{Emma’s monthly income must be at least}} \\ {\text{First write a sentence that gives the information}} &{\text{three times the rent.}} \\ {\text{to find it.}} &{} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{5265 \geq 3r} \\ {\text{Remember, } a > x\text{ has the same meaning}} &{1755 \geq r} \\ {\text{as }x < a} &{r \leq 1755} \\ {\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{A maximum rent of \$1,755 seems}} &{} \\ {\text{reasonable for an income of \$5,265.}} &{} \\ {\textbf{Step 7. Answer} \text{ the answer in the problem}} &{\text{the question with a}} \\ {\text{complete sentence.}} &{\text{The maximum rent is \$1,755.}} \end{array}\)

Zoezi\(\PageIndex{2}\)

Alan anapakia godoro na masanduku ambayo kila mmoja hupima paundi 45. Pallet inaweza kusaidia salama zaidi ya paundi 900. Ni masanduku ngapi anayeweza kupakia salama kwenye pala?

Jibu

Hatuwezi kuwa na masanduku zaidi ya 20.

Zoezi\(\PageIndex{3}\)

Lifti katika jengo la ghorofa la Yehire ina ishara inayosema uzito wa kiwango cha juu ni paundi 2,100. Ikiwa uzito wa mtu mmoja ni paundi 150, ni watu wangapi wanaoweza kupanda lifti salama?

Jibu

Upeo wa watu 14 wanaweza kupanda salama katika lifti.

Wakati mwingine maombi inahitaji ufumbuzi kuwa namba nzima, lakini suluhisho la algebraic kwa usawa sio namba nzima. Katika hali hiyo, tunapaswa kuzunguka suluhisho la algebraic kwa idadi nzima. muktadha wa maombi itaamua kama sisi pande zote juu au chini. Kuangalia maombi kama hii, sisi pande zote jibu letu kwa idadi ambayo ni rahisi kukokotoa na kuhakikisha kwamba idadi inafanya usawa kweli.

Zoezi\(\PageIndex{4}\)

Dawn alishinda ruzuku mini-ya $4,000 kununua kompyuta kibao kwa ajili ya darasa lake. Vidonge ambavyo angependa kununua gharama $254.12 kila mmoja, ikiwa ni pamoja na kodi na utoaji. Nambari ya juu ya vidonge Dawn inaweza kununua nini?

Jibu

\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the maximum number of tablets Dawn can buy}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let n = the number of tablets.}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{} \\{\text{gives the information to find it.}} &{$254.12\text{ times the number of tablets is no}} \\ {} &{\text{more than \$4000.}} \\ {\text{Translate into an inequality.}} &{254.12n \leq 4000} \\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{n \leq 15.74} \\ {\text{But n must be a whole number of tablets,}} &{} \\ {\text{so round to 15.}} &{n \leq 15}\\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{Rounding down the price to \$250,}} &{} \\ {\text{15 tablets would cost \$3750, while}} &{} \\ {\text{16 tablets would be \$4000. So a}} &{} \\{\text{maximum of 15 tablets at \$254.12}} &{} \\ {\text{seems reasonable.}} &{} \\{\textbf{Step 7. Answer} \text{ the answer in the problem}} &{\text{the question with a}} \\ {\text{complete sentence.}} &{\text{Dawn can buy a maximum of 15 tablets.}} \end{array}\)

Zoezi\(\PageIndex{5}\)

Angie ina $20 kutumia kwenye masanduku ya juisi kwa picnic ya mapema ya mtoto wake. Kila pakiti ya masanduku ya juisi hupunguza $2.63. Nambari ya juu ya pakiti ambazo anaweza kununua ni nini?

Jibu

pakiti saba

Zoezi\(\PageIndex{6}\)

Daniel anataka kumshangaa mpenzi wake na chama cha kuzaliwa kwenye mgahawa wake unaopenda. Itakuwa gharama $42.75 kwa kila mtu kwa chakula cha jioni, ikiwa ni pamoja na ncha na kodi. Bajeti yake kwa ajili ya chama ni $500. Idadi ya juu ya watu Daniel anaweza kuwa nayo katika chama gani?

Jibu

watu 11

Zoezi\(\PageIndex{7}\)

Pete anafanya kazi kwenye duka la kompyuta. Malipo yake ya kila wiki yatakuwa ama kiasi cha kudumu, $925, au $500 pamoja na 12% ya mauzo yake ya jumla. Je, mauzo yake yote yanapaswa kuwa kiasi gani kwa chaguo lake la kulipa kutofautiana ili kuzidi kiasi cha kudumu cha $925?

Jibu

\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the total sales needed for his variable pay}} \\ {} &{\text{option to exceed the fixed amount of \$925}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let s = the total sales.}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{} \\{\text{gives the information to find it.}} &{$500\text{ plus 12% of total sales is more than \$925.}} \\ {\text{Translate into an inequality. Remember to}} &{500 + 0.12s > 925} \\{\text{convert the percent to a decimal.}} &{} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{0.12s > 425} \\ {} &{s > 3541.\overline{66}} \\ \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{Rounding down the price to \$250,}} &{} \\ {\text{15 tablets would cost \$3750, while}} &{} \\ {\text{If we round the total sales up to}} &{} \\{\text{\$4000, we see that}} &{} \\ {\text{500+0.12(4000) = 980, which is more}} &{} \\ {\text{than \$925.}} &{} \\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{The total sales must be more than \$3541.67}} \end{array}\)

Zoezi\(\PageIndex{8}\)

Tiffany tu alihitimu kutoka chuo na kazi yake mpya itamlipa $20000 kwa mwaka pamoja na 2% ya mauzo yote. Anataka kupata angalau $100000 kwa mwaka. Kwa mauzo gani atakuwa na uwezo wa kufikia lengo lake?

Jibu

angalau $4000000

Zoezi\(\PageIndex{9}\)

Christian amepewa kazi mpya ambayo hulipa $24000 mwaka pamoja na 3% ya mauzo. Kwa nini jumla ya mauzo bila kazi hii mpya kulipa zaidi ya kazi yake ya sasa ambayo inalipa $60000?

Jibu

angalau $1200000

Zoezi\(\PageIndex{10}\)

Sergio na Lizeth wana bajeti ya likizo kali sana. Wanapanga kukodisha gari kutoka kampuni inayoshutumu $75 kwa wiki pamoja na $0.25 kwa maili. Jinsi maili wengi wanaweza kusafiri na bado kuweka ndani yao $200 bajeti?

Jibu

\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the number of miles Sergio and Lizeth can travel}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {} &{\text{Let m = the number of miles.}} \\ {\text{Choose a variable to represent that quantity.}} &{} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{\text{\$75 plus 0.25 times the number of miles is}} \\{\text{gives the information to find it.}} &{\text{ less than or equal to \$200.}} \\ {\text{Translate into an inequality. }} &{75 + 25m \leq 200} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{0.25m \leq 125} \\ {} &{m \leq 500 \text{ miles}} \\ \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{Yes, 75 + 0.25(500) = 200.}} & {}\\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{Sergio and Lizeth can travel 500 miles}} \\ {} &{\text{and still stay on budget.}} \end{array}\)

Zoezi\(\PageIndex{11}\)

Mpango wa simu wa Taleisha unamgharimu $28.80 kwa mwezi pamoja na $0.20 kwa kila ujumbe wa maandishi. Ujumbe wangapi wa maandishi anaweza kutumia na kuweka muswada wake wa simu ya kila mwezi si zaidi ya $50?

Jibu

si zaidi ya 106 ujumbe wa maandishi

Zoezi\(\PageIndex{12}\)

Muswada wa joto wa Rameen ni $5.42 kwa mwezi pamoja na $1.08 kwa therm. Ngapi therms unaweza Rameen kutumia kama anataka muswada wake inapokanzwa kuwa kiwango cha juu cha $87.50?

Jibu

si zaidi ya 76 therms

Lengo la kawaida la biashara nyingi ni kupata faida. Faida ni pesa iliyobaki wakati gharama zimeondolewa kutoka pesa zilizopatikana. Katika mfano unaofuata, tutapata idadi ya ajira mfanyabiashara mdogo anahitaji kufanya kila mwezi ili kufanya kiasi fulani cha faida.

Zoezi\(\PageIndex{13}\)

Elliot ina mazingira ya matengenezo ya biashara. Gharama zake za kila mwezi ni $1,100. Ikiwa anadai $60 kwa kazi, ni ajira ngapi lazima afanye ili kupata faida ya angalau $4,000 kwa mwezi?

Jibu

\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the number of jobs Elliot needs}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {\text{Choose a variable to represent it}} &{\text{Let j = the number of jobs.}} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{\text{\$60 times the number of jobs minus \$1,100 is at least \$4,000.}} \\{\text{gives the information to find it.}} &{\text{ less than or equal to \$200.}} \\ {\text{Translate into an inequality. }} &{60j - 1100 \geq 4000} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{60j \geq 5100} \\ {} &{j \geq 85\text{ jobs}} \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{If Elliot did 90 jobs, his profit would be}} & {}\\ {\text{60(90)−1,100,or \$4,300. This is}} &{} \\ {\text{more than \$4,000.}} &{} \\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{Elliot must work at least 85 jobs.}} \end{array}\)

Zoezi\(\PageIndex{14}\)

Caleb ana mnyama ameketi biashara. Yeye mashtaka $32 kwa saa. Gharama zake za kila mwezi ni $2272. Ni masaa ngapi anapaswa kufanya kazi ili kupata faida ya angalau $800 kwa mwezi?

Jibu

angalau masaa 96

Zoezi\(\PageIndex{15}\)

Felicity ina biashara ya calligraphy. Yeye mashtaka $2.50 kwa mwaliko harusi. Gharama zake za kila mwezi ni $650. Ni mialiko ngapi lazima aandike ili kupata faida ya angalau $2800 kwa mwezi?

Jibu

angalau mialiko 1380

Wakati mwingine maisha hupata ngumu! Kuna hali nyingi ambazo kiasi kadhaa huchangia gharama zote. Lazima tuhakikishe akaunti kwa gharama zote za mtu binafsi wakati tunatatua matatizo kama haya.

Zoezi\(\PageIndex{16}\)

Rafiki bora wa Brenda ni kuwa na harusi ya marudio na tukio hilo litaendelea siku 3. Brenda ina $500 katika akiba na wanaweza kupata $15 saa babysitting. Anatarajia kulipa dola 350 za ndege, $375 kwa ajili ya chakula na burudani na $60 usiku kwa sehemu yake ya chumba cha hoteli. Ni saa ngapi lazima yeye babysit kuwa na fedha za kutosha kulipa kwa ajili ya safari?

Jibu

\(\begin{array} {ll} {\textbf{Step 1. Read} \text{ the problem.}} &{} \\ {\textbf{Step 2. Identify} \text{ what we are looking for.}} &{\text{the number of hours Brenda must babysit}} \\ {\textbf{Step 3. Name} \text{ what we are looking for.}} &{} \\ {\text{Choose a variable to represent that quantity.}} &{\text{Let h = the number of hours.}} \\{\textbf{Step 4. Translate.} \text{ write a sentence that}} &{} \\{\text{gives the information to find it.}} &{} \\ {} &{\text{The expenses must be less than or equal to}} \\ {} &{\text{the income. The cost of airfare plus the}} \\ {} &{\text{cost of food and entertainment and the}} \\ {} &{\text{hotel bill must be less than or equal to the savings}} \\ {} &{\text{plus the amount earned babysitting.}} \\ {\text{Translate into an inequality. }} &{\$350 + \$375 + \$60(3) \leq \$500 + \$15h} \\\\ {\textbf{Step 5. Solve} \text{ the inequality.}} &{905 \leq 500 + 15h} \\{} &{405 \leq 15h} \\ {} &{27 \leq h} \\ {} &{h \geq 27} \\ \\{\textbf{Step 6. Check} \text{ the answer in the problem}} &{} \\ {\text{and make sure it makes sense.}} &{} \\ {\text{We substitute 27 into the inequality.}} & {}\\{905 \leq 500 + 15h} &{} \\ {905 \leq 500 + 15(27)} &{} \\ {905 \leq 905} &{} \\ \\{\textbf{Step 7. Answer} \text{ the the question with a complete sentence.}} &{\text{Brenda must babysit at least 27 hours.}} \end{array}\)

Zoezi\(\PageIndex{17}\)

Malik anapanga safari ya likizo ya majira ya joto ya siku 6. ana $840 katika akiba, naye chuma $45 kwa saa kwa ajili ya Tutoring. Safari hiyo itamlipa $525 kwa ndege, $780 kwa chakula na kuona, na $95 kwa usiku kwa hoteli. Ni saa ngapi anapaswa kufundisha kuwa na pesa za kutosha kulipa safari?

Jibu

angalau masaa 23

Zoezi\(\PageIndex{18}\)

Josue anataka kwenda safari ya siku 10 ya barabara ijayo spring. Itakuwa gharama yake $180 kwa ajili ya gesi, $450 kwa ajili ya chakula, na $49 kwa usiku kwa motel. ana $520 katika akiba na wanaweza kupata $30 kwa driveway shoveling theluji. Jinsi driveways wengi lazima yeye koleo kuwa na fedha za kutosha kulipa kwa ajili ya safari?

Jibu

angalau 20 driveways

Dhana muhimu

• Kutatua kutofautiana
  1. Soma tatizo.
  2. Tambua kile tunachotafuta.
  3. Jina kile tunachotafuta. Chagua variable kuwakilisha kiasi hicho.
  4. Tafsiri. Andika sentensi ambayo inatoa taarifa ili kuipata. Tafsiri katika usawa.
  5. Kutatua kukosekana kwa usawa.
  6. Angalia jibu katika tatizo na uhakikishe kuwa ni busara.
  7. Jibu swali kwa sentensi kamili.