Whakaoti i te 5-2x/3+4<3x-5/4*3x/2

$$\frac { 5 - 2 x } { 3 } + 4 < \frac { 3 x - 5 } { 4 } \cdot \frac { 3 x } { 2 }$$

Whakaoti mō x

$$x\in \left(-\infty,\frac{29-\sqrt{15529}}{54}\right)\cup \left(\frac{\sqrt{15529}+29}{54},\infty\right)$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 12, arā, te tauraro pātahi he tino iti rawa te kitea o 3,4,2. I te mea he tōrunga te 12, kāore e huri te ahunga koreōrite.

$$4\left(5-2x\right)+48<3\left(3x-5\right)\times \frac{3x}{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 5-2x.

$$20-8x+48<3\left(3x-5\right)\times \frac{3x}{2}$$

Tāpirihia te 20 ki te 48, ka 68.

$$68-8x<3\left(3x-5\right)\times \frac{3x}{2}$$

Tuhia te 3\times \frac{3x}{2} hei hautanga kotahi.

$$68-8x<\frac{3\times 3x}{2}\left(3x-5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{3\times 3x}{2} ki te 3x-5.

$$68-8x<3\times \frac{x\times 3^{2}}{2}x-5\times \frac{3\times 3x}{2}$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$68-8x<3\times \frac{x\times 9}{2}x-5\times \frac{3\times 3x}{2}$$

Tuhia te 3\times \frac{x\times 9}{2} hei hautanga kotahi.

$$68-8x<\frac{3x\times 9}{2}x-5\times \frac{3\times 3x}{2}$$

Tuhia te \frac{3x\times 9}{2}x hei hautanga kotahi.

$$68-8x<\frac{3x\times 9x}{2}-5\times \frac{3\times 3x}{2}$$

Whakareatia te 3 ki te 3, ka 9.

$$68-8x<\frac{3x\times 9x}{2}-5\times \frac{9x}{2}$$

Tuhia te -5\times \frac{9x}{2} hei hautanga kotahi.

$$68-8x<\frac{3x\times 9x}{2}+\frac{-5\times 9x}{2}$$

Tā te mea he rite te tauraro o \frac{3x\times 9x}{2} me \frac{-5\times 9x}{2}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$68-8x<\frac{3x\times 9x-5\times 9x}{2}$$

Mahia ngā whakarea i roto o 3x\times 9x-5\times 9x.

$$68-8x<\frac{27x^{2}-45x}{2}$$

Whakawehea ia wā o 27x^{2}-45x ki te 2, kia riro ko \frac{27}{2}x^{2}-\frac{45}{2}x.

$$68-8x<\frac{27}{2}x^{2}-\frac{45}{2}x$$

Tangohia te \frac{27}{2}x^{2} mai i ngā taha e rua.

$$68-8x-\frac{27}{2}x^{2}<-\frac{45}{2}x$$

Me tāpiri te \frac{45}{2}x ki ngā taha e rua.

$$68-8x-\frac{27}{2}x^{2}+\frac{45}{2}x<0$$

Pahekotia te -8x me \frac{45}{2}x, ka \frac{29}{2}x.

$$68+\frac{29}{2}x-\frac{27}{2}x^{2}<0$$

Me whakarea te koreōrite ki te -1 kia tōrunga ai te tau whakarea o te pū tino teitei i 68+\frac{29}{2}x-\frac{27}{2}x^{2}. I te mea he tōraro a -1, ka huri te ahunga koreōrite.

$$-68-\frac{29}{2}x+\frac{27}{2}x^{2}>0$$

Kia whakaotia te koreōrite, me tauwehe te taha mauī. Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$-68-\frac{29}{2}x+\frac{27}{2}x^{2}=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te \frac{27}{2} mō te a, te -\frac{29}{2} mō te b, me te -68 mō te c i te ture pūrua.

$$x=\frac{-\left(-\frac{29}{2}\right)±\sqrt{\left(-\frac{29}{2}\right)^{2}-4\times \frac{27}{2}\left(-68\right)}}{2\times \frac{27}{2}}$$

Mahia ngā tātaitai.

$$x=\frac{\frac{29}{2}±\frac{1}{2}\sqrt{15529}}{27}$$

Whakaotia te whārite x=\frac{\frac{29}{2}±\frac{1}{2}\sqrt{15529}}{27} ina he tōrunga te ±, ina he tōraro te ±.

$$x=\frac{\sqrt{15529}+29}{54} x=\frac{29-\sqrt{15529}}{54}$$

Tuhia anō te koreōrite mā te whakamahi i ngā otinga i whiwhi.

$$\frac{27}{2}\left(x-\frac{\sqrt{15529}+29}{54}\right)\left(x-\frac{29-\sqrt{15529}}{54}\right)>0$$

Kia tōrunga te otinga, me tōraro tahi te x-\frac{\sqrt{15529}+29}{54} me te x-\frac{29-\sqrt{15529}}{54}, me tōrunga tahi rānei. Whakaarohia te tauira ina he tōraro tahi te x-\frac{\sqrt{15529}+29}{54} me te x-\frac{29-\sqrt{15529}}{54}.

$$x-\frac{\sqrt{15529}+29}{54}<0 x-\frac{29-\sqrt{15529}}{54}<0$$

Te otinga e whakaea i ngā koreōrite e rua ko x<\frac{29-\sqrt{15529}}{54}.

$$x<\frac{29-\sqrt{15529}}{54}$$

Whakaarohia te tauira ina he tōrunga tahi te x-\frac{\sqrt{15529}+29}{54} me te x-\frac{29-\sqrt{15529}}{54}.

$$x-\frac{29-\sqrt{15529}}{54}>0 x-\frac{\sqrt{15529}+29}{54}>0$$

Te otinga e whakaea i ngā koreōrite e rua ko x>\frac{\sqrt{15529}+29}{54}.

$$x>\frac{\sqrt{15529}+29}{54}$$

Ko te otinga whakamutunga ko te whakakotahi i ngā otinga kua whiwhi.

$$x<\frac{29-\sqrt{15529}}{54}\text{; }x>\frac{\sqrt{15529}+29}{54}$$