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10.2: Kutatua na Kubadilisha kutofautiana, na Kuandika Majibu katika Uthibitishaji wa Muda

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    164789
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    Ili kutatua na kutofautiana kwa grafu:

    1. Tatua usawa kwa kutumia Mali ya Usawa kutoka sehemu iliyopita.
    2. Grafu suluhisho lililowekwa kwenye mstari wa nambari.
    3. Andika suluhisho lililowekwa katika notation ya muda.

    Tatua usawa, graph ufumbuzi uliowekwa kwenye mstari wa nambari na uonyeshe suluhisho lililowekwa katika maelezo ya muda:

    1. \(−1 ≤ 2x − 5 < 7\)
    2. \(x^2 + 7x + 10 < 0\)
    3. \(−6 < x − 2 < 4\)
    Suluhisho
    1. \(\begin{array} &&−1 ≤ 2x − 5 < 7 &\text{Example problem} \\ &−1 + 5 ≤ 2x − 5 + 5 < 7 + 5 &\text{The goal is to isolate the variable \(x\), hivyo kuanza kwa kuongeza\(5\) mikoa yote mitatu katika usawa.}\\ &4 ≤ 2x <12 &\ maandishi {Kurahisisha.}\\ &\ dfrac {4} {2} ≤ 2x^2 <\ dfrac {4} {2} {2} &\ Nakala {Gawanya yote kwa\(2\) kutenganisha variable\(x\).}\\ &2 ≤ x <6 &\ Nakala {Jibu la mwisho lililoandikwa ukosefu wa uhaba/ufumbuzi kuweka fomu.}\\ & [2, 6) &\ Nakala {Jibu la mwisho lililoandikwa katika nukuu ya muda (angalia sehemu ya Nukuu ya Muda kwa maelezo zaidi)}\ mwisho {safu}\)

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    1. \(\begin{array} &&x^2 + 7x + 10 < 0 &\text{Example problem} \\ &(x + 5)(x + 2) < 0 &\text{Factor the polynomial.} \\ &(x + 5)(x + 2) < 0 &\text{The product must be less than \(0\), ambayo ina maana kwamba kama\((x + 5) > 0\), basi\((x + 2) < 0\). Vile vile\((x + 5) < 0\), ikiwa, basi\((x + 2) > 0\).}\\ & (x + 5) > 0 (x + 2) < 0 &\ text {Find the intersection of each of these inequalities.}\\ &x > -5 x <-2 &\ maandishi {Pata makutano ya kila moja ya usawa huu.} \ mwisho {safu}\)
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    Suluhisho lililowekwa kwa\(x > −5\).
    clipboard_e9ed6522e3f219b2195a81e9a9f3a9253.png
    Suluhisho lililowekwa kwa\(x − 2\).

    \(\begin{array} &&\;\;\;−5 < x < −2 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Final answer written in inequality/solution set form.} \\ &\;\;\;(−5, −2) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{Final answer written in interval notation (see section on Interval Notation for more details).} \end{array}\)

    1. \(\begin{array}&&−6 < x − 2 ≤ 4 &\text{Example problem} \\ &−6 + 2 < x − 2 + 2 ≤ 4 + 2 &\text{The goal is to isolate the variable \(x\), hivyo kuanza\(2\) kwa kuongeza mikoa yote mitatu katika usawa.}\\ &-4 <x ≤ 6 &\ maandishi {Jibu la mwisho lililoandikwa katika fomu ya kutosawa/ufumbuzi uliowekwa.}\\ & (-4, 6] &\ maandishi {Jibu la mwisho lililoandikwa katika nukuu ya muda (angalia sehemu ya Nukuu ya Muda kwa maelezo zaidi).} \ mwisho {safu}\)

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    Tatua kutofautiana, graph seti ya suluhisho kwenye mstari wa nambari na uonyeshe seti ya suluhisho katika nukuu ya muda:

    1. \(0 ≤ x + 1 ≤ 4\)
    2. \(0 < 2(x − 1) ≤ 4\)
    3. \(6 < 2(x − 1) < 12\)
    4. \(x^2 − 6x − 16 < 0\)
    5. \(2x^2 − x − 15 > 0\)