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6.2: Kutatua Ulinganisho wa Thamani kamili

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    164693
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    Ili kutatua usawa wa thamani kamili, kwanza fikiria mali mbili zifuatazo za thamani kamili:

    Ufafanuzi: Mali ya Thamani kamili

    mali 1: Kwa\(b > 0\),\(|a| = b\) kama na tu kama\(a = b\) au\(a = −b\)

    Mali 2: Kwa idadi yoyote halisi\(a\) na\(b\),\(|a| = |b|\) kama na tu kama\(a = b\) au\(a = −b\)

    • Kabla ya Mali 1 inatumiwa, jitenga kujieleza kwa thamani kamili kwa upande wowote wa equation.
    • Angalia ufumbuzi kwa kuwabadilisha tena kwenye equation ya awali.
    • Ufumbuzi huwasilishwa kama seti ya suluhisho la fomu\(\{p, q\}\), wapi\(p\) na\(q\) ni idadi yoyote halisi.
    • Seti ya suluhisho ya equation ya thamani kamili imewekwa kama pointi kwenye mstari wa nambari.

    Tatua kila equation na graph kuweka ufumbuzi.

    1. \(|x| = 7\)
    2. \(|5x – 3| = 2\)
    3. \(|20 – x| = −80\)

    Suluhisho

    1. Ili kutatua\(|x| = 7\), tumia Mali 1\(a = x\) na\(b = 7\).

    Kwa hiyo, ufumbuzi ni,\(x = −7\) na\(x = 7\), na kuweka suluhisho ni\(\{-7,7\}\). Grafu ya kuweka suluhisho ni kama inavyoonekana katika takwimu hapa chini.

    clipboard_e4e661622afb19f733118b1049d678a57.png

    1. Njia ya kutatua equation kutumika katika sehemu a inaweza kupanuliwa kwa equation iliyotolewa katika sehemu hii na\(a = 5x – 3\) na\(b = 2\).

    Hivyo, equation ya thamani kabisa\(|5x – 3| = 2\) ni sawa na:

    \(\begin{array} &&5x − 3 = 2 &\text{ or } &5x − 3 = −2 &\text{Property 1} \\ &5x = 5 &\text{ or } &5x = 1 &\text{Add \(3\)kwa pande zote mbili za equations}\\ &x = 1 &\ maandishi {au} &x =\ dfrac {1} {5} &\ text {Gawanya kwa pande\(5\) zote mbili za equations}\ mwisho {array}\)

    Sasa, angalia ikiwa\(x = 1\) na\(x = \dfrac{1}{5}\) ni ufumbuzi wa usawa wa thamani kamili.

    \(\begin{array} &&\text{For } x = 1 &\text{For } x = \dfrac{1}{5} &\\ &|5x − 3| = 2 &|5x − 3| = 2 &\text{Given} \\ &|5(1) − 3| \stackrel{?}{=} 2 &|5 \left( \dfrac{1}{5} \right) − 3| \stackrel{?}{=} 2 &\text{Substitute the \(x\)-maadili}\\ &|5 - 3|\ stackrel {?} {=} 2 &|1 - 3|\ stackrel {?} {=} 2 &\ maandishi {Kurahisisha}\\ &|2|\ stackrel {?} {=} 2 &||2|\ stackrel {?} {=} 2 &\ maandishi {Tumia ufafanuzi wa thamani kamili}\\ &2 = 2\;\ alama &2 = 2\;\ checkmark\ mwisho {safu}\)

    Kwa kuwa equations hapo juu ni kweli, basi,\(x = 1\) na\(x = \dfrac{1}{5}\) ni ufumbuzi wa equation kamili ya thamani. Suluhisho la kuweka ni\(\left\{\dfrac{1}{5} , 1\right\}\). Grafu ya kuweka suluhisho ni kama inavyoonekana katika takwimu hapa chini.

    clipboard_e78f4a9bbcf248134874230631b7dada3.png

    1. Kwa kuwa thamani kamili haiwezi kamwe kuwa hasi, hakuna namba halisi\(x\) inayofanya\(|20 – x| = −80\) kweli. equation haina ufumbuzi na ufumbuzi kuweka ni\(∅\).

    Tatua na graph kuweka suluhisho.

    1. \(\left| \dfrac{4}{3} x + 3 \right| + 8 = 18\)
    2. \(4 \left| \dfrac{1}{3}x − 6 \right| − 5 = −5\)
    3. \(|4x – 3| = |x + 6|\)

    Suluhisho

    1. Angalia kwamba usemi wa thamani kamili haujatengwa ambayo inamaanisha mali haiwezi kutumika. Kwanza, jitenga\(\left| \dfrac{4}{3}x + 3 \right|\) upande wa kushoto wa equation, basi, tumia Mali 1.

    \(\begin{array} &&\left| \dfrac{4}{3} x + 3 \right| + 8 = 18 &\text{Given equation} \\ & \left| \dfrac{4}{3} + 3 \right| = 10 &\text{Subtract \(8\)kutoka pande zote mbili za equation}\ mwisho {array}\)

    Kwa thamani kamili sasa pekee, kutatua\(\left| \dfrac{4}{3} + 3 \right| = 10\) kutumia Mali 1,\(a = \dfrac{4}{3} x + 3\) na\(b = 10\) kama ifuatavyo,

    \(\begin{array} && &\left| \dfrac{4}{3} + 3 \right| = 10 & & \\ &\dfrac{4}{3} + 3 = 10 &\text{ or } & \dfrac{4}{3} + 3 = -10 &\text{Property 1} \\ &\dfrac{4}{3} x = 7 &\text{ or } &\dfrac{4}{3}x = −13 &\text{Subtract \(3\)kutoka pande zote mbili}\\ &x =\ dfrac {21} {4} &\ maandishi {au} &x = -\ dfrac {39} {4} &\ maandishi {Kuzidisha pande zote mbili kwa\(\dfrac{3}{4}\)}\ mwisho {safu}\)

    Angalia ufumbuzi\(x = −\dfrac{39}{4}\) na\(x = \dfrac{21}{4}\) kwa kuwabadilisha katika equation ya awali ya thamani kamili. Suluhisho lililowekwa ni\(\left\{ −\dfrac{39}{4}, \dfrac{21}{4} \right\}\) na grafu ya kuweka suluhisho ni kama inavyoonekana katika takwimu hapa chini.

    clipboard_e42b1bbc90f71c52e8a95664a185e2c67.png

    1. Sawa na sehemu a, jitenga kujieleza thamani kamili. Hivyo, kwanza kujitenga\(\left| \dfrac{1}{3} x − 6 \right|\) upande wa kushoto wa equation na kuomba Mali 1.

    \(\begin{array} &&4 \left| \dfrac{1}{3}x − 6 \right| − 5 = −5 &\text{Given equation} \\ &4 \left| \dfrac{1}{3}x − 6 \right| = 0 &\text{Add \(5\)kwa pande zote mbili za equation}\\ &\ kushoto|\ dfrac {1} {3} x - 6\ haki| = 0 &\ maandishi {Gawanya kwa pande\(4\) zote mbili za equation}\ mwisho {array}\)

    Thamani kamili ni pekee. Kwa kuwa\(0\) ni namba pekee ambayo thamani kamili ni\(0\), maneno\(\dfrac{1}{3}x − 6\) lazima iwe sawa na\(0\). Hivyo,

    \(\begin{array} &&\dfrac{1}{3}x − 6 = 0 & \\ &\dfrac{1}{3}x − 6 &\text{Add \(6\)kwa pande zote mbili za equation}\\ &x = 18 &\ Nakala {Kuzidisha pande zote mbili kwa\(3\)}\ mwisho {array}\)

    Suluhisho ni\(18\) na kuweka suluhisho ni\(\{18\}\). Thibitisha kwamba inatimiza equation ya awali. Grafu ya kuweka suluhisho ni kama inavyoonekana katika takwimu hapa chini.

    clipboard_e2acd5153df84bfef5569962926db627b.png

    1. \(|4x − 7| = |x + 14|\)Kumbuka kwamba kutatua\(|4x − 7| = |x + 14|\), kutumia Mali 2 na\(a = 4x − 7\) na\(b = x + 14\).

    \(\begin{array} && &|4x − 7| = |x + 14| & &\text{Given} \\ &4x−7 = x+14 &\text{ or } &4x − 7 = −(x + 14) &\text{Property 2} \\ &4x−7 = x+14 &\text{ or } &4x − 7 = −x − 14 &\text{Distribute \(−1\)ili kurahisisha equation sahihi}\\ &4x = x + 21 &\ maandishi {au} &4x = -x - 7 &\ maandishi {Ongeza\(7\) pande zote mbili za usawa}\\ &3x = 21 &\ maandishi {au} &5x = -7 &\ maandishi {Kurahisisha}\\ &x = 7 &\ Nakala {au} &x = ∙\ dfrac {7} &\ Nakala {Gawanya kila equation na ya\(x\) -mgawo}\ mwisho {array}\)

    Angalia ufumbuzi\(x = −\dfrac{7}{5}\) na\(x = 7\) kwa kuwabadilisha katika equation ya awali ya thamani kamili. Suluhisho la kuweka ni\(\left\{ −\dfrac{7}{5}, 7\right\}\). Grafu ya suluhisho ni kama inavyoonekana katika takwimu hapa chini.

    clipboard_eb0f0ab26e578678046e463bf1d8ac854.png

    Tatua kila equation, angalia suluhisho na graph kuweka suluhisho.

    1. \(|x| = 19\)
    2. \(|x − 4| = 10\)
    3. \(|2x − 5| = 12\)
    4. \(\left|\dfrac{x}{11} \right| = 2.5\)
    5. \(|x − 3.8| = −2.7\)
    6. \(|3x − 4.5| = 9.3\)
    7. \(\dfrac{8}{3} |x − 6| = 14\)
    8. \(|x + 15| − 19 = 7\)
    9. \(|11x + 3| + 28 = 16\)
    10. \( \left| \dfrac{8}{7} x + 9 \right| − 2 = 8\)
    11. \( −3|2x − 7| + 13 = 13\)
    12. \( 8 − 5|10x + 6| = 5\)
    13. \( |5x − 14| = |3x − 9|\)
    14. \( |15x| = |x − 21|\)
    15. \( |4x − 7| = |5(2x + 3)|\)
    16. \( \dfrac{7}{8} = \dfrac{3x}{2} + \dfrac{2x}{5}\)