5-2x/3+4<3x-5/4*3x/2 eching

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

x uchun yechish

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

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 12 ga, 3,4,2 ning eng kichik karralisiga ko‘paytiring. 12 musbat bo‘lgani uchun, tengsizlik yo‘nalishi o‘zgarmaydi.

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

4 ga 5-2x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

68 olish uchun 20 va 48'ni qo'shing.

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

3\times \frac{3x}{2} ni yagona kasrga aylantiring.

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

\frac{3\times 3x}{2} ga 3x-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

2 daraja ko‘rsatkichini 3 ga hisoblang va 9 ni qiymatni oling.

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

3\times \frac{x\times 9}{2} ni yagona kasrga aylantiring.

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

\frac{3x\times 9}{2}x ni yagona kasrga aylantiring.

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

9 hosil qilish uchun 3 va 3 ni ko'paytirish.

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

-5\times \frac{9x}{2} ni yagona kasrga aylantiring.

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

\frac{3x\times 9x}{2} va \frac{-5\times 9x}{2} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

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

3x\times 9x-5\times 9x ichidagi ko‘paytirishlarni bajaring.

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

\frac{27}{2}x^{2}-\frac{45}{2}x natijani olish uchun 27x^{2}-45x ning har bir ifodasini 2 ga bo‘ling.

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

Ikkala tarafdan \frac{27}{2}x^{2} ni ayirish.

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

\frac{45}{2}x ni ikki tarafga qo’shing.

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

\frac{29}{2}x ni olish uchun -8x va \frac{45}{2}x ni birlashtirish.

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

68+\frac{29}{2}x-\frac{27}{2}x^{2} musbatida eng katta quvvatni koeffitsientini aniqlash uchun tengsizlikni -1 ga koʻpaytiring. -1 manfiy boʻlgani uchun tengsizlikning yo‘nalishi o‘zgaradi.

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

Tengsizlikni yechish uchun chap tomon faktorini hisoblang. Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

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

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni bu formula bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat tenglamada a uchun \frac{27}{2} ni, b uchun -\frac{29}{2} ni va c uchun -68 ni ayiring.

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

Hisoblarni amalga oshiring.

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

x=\frac{\frac{29}{2}±\frac{1}{2}\sqrt{15529}}{27} tenglamasini ± plus va ± minus boʻlgan holatida ishlang.

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

Yechimlardan foydalanib tengsizlikni qaytadan yozing.

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

Koʻpaytma musbat boʻlishi uchun x-\frac{\sqrt{15529}+29}{54} va x-\frac{29-\sqrt{15529}}{54} ikkalasi yo manfiy, yo musbat boʻlishi kerak. x-\frac{\sqrt{15529}+29}{54} va x-\frac{29-\sqrt{15529}}{54} ikkalasi manfiy boʻlganda, yechimini toping.

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

Ikkala tengsizlikning mos yechimi – x<\frac{29-\sqrt{15529}}{54}.

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

x-\frac{\sqrt{15529}+29}{54} va x-\frac{29-\sqrt{15529}}{54} ikkalasi musbat boʻlganda, yechimini toping.

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

Ikkala tengsizlikning mos yechimi – x>\frac{\sqrt{15529}+29}{54}.

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

Oxirgi yechim olingan yechimlarning birlashmasidir.

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