Бүтін рационал теңсіздіктер (Рустюмова 1.5.2А 1-30)

Шешуі:

$${11 - \left( {{x^2} + 2x + 1} \right) - x \ge 0}$$ $${11 - {x^2} - 2x - 1 - x \ge 0}$$ $${ - {x^2} - 3x + 10 \ge 0}$$ $${ - (x + 5)(x - 2) \ge 0}$$

$${(x + 5)(x - 2) \le 0}$$ $${x + 5 = 0,\quad x - 2 = 0;}$$ $${x = - 5,\quad x = 2}$$

Жауабы: $x \in[-5 ; 2]$

Шешуі:

$${(2x - 8)(2x - 8 - 4x) \ge 0}$$ $${(2x - 8)( - 2x - 8) \ge 0}$$ $${2 \cdot ( - 2) \cdot (x - 4)(x + 4) \ge 0}$$

$${ - 4(x - 4)(x + 4) \ge 0}$$ $${ - 4(x - 4)(x + 4) = 0}$$ $${x - 4 = 0,\quad x + 4 = 0}$$ $${x = 4,\quad x = - 4}$$

Жауабы: $x \in[-4 ; 4]$

Шешуі:

$${{x^2} + 5x - 2 \gt 4x}$$ $${{x^2} + x - 2 \gt 0}$$

$${(x + 2)(x - 1) \gt 0}$$ $$x = - 2;\quad x = 1$$

Жауабы: $(-\infty ;-2) \cup(1 ;+\infty)$

Шешуі:

$${{x^2} + 9x + 18 \lt 0}$$ $${(x + 6)(x + 3) \lt 0}$$

$${(x + 6)(x + 3) = 0}$$ $${x = - 6,\quad x = - 3}$$

Жауабы: $x \in(-6 ;-3)$

Шешуі:

$${x - \frac{{{x^2}}}{2} + 4x - \frac{{11}}{2} \gt 0}$$ $${ - \frac{{{x^2}}}{2} + 5x - \frac{{11}}{2} \gt 0}$$ $${ - {x^2} + 10x - 11 \gt 0}$$ $${ - {x^2} + 10x - 11 = 0}$$ $${D = 100 - 44 = 56}$$

$${{x_1} = \frac{{ - 10 - \sqrt {56} }}{{ - 2}} = \frac{{ - 10 - 2\sqrt {14} }}{{ - 2}} = 5 + \sqrt {14} }$$ $${{x_2} = \frac{{ - 10 + \sqrt {56} }}{{ - 2}} = 5 - \sqrt {14} }$$

Жауабы: $x \in(5-\sqrt{14} ; 5+\sqrt{14})$

Шешуі:

$${2x - {x^2} - 1 \lt 0}$$ $${ - {x^2} + 2x - 1 \lt 0}$$ $${ - \left( {{x^2} - 2x + 1} \right) \lt 0}$$

$${ - {{(x - 1)}^2} \lt 0}$$ $${x = 1}$$

Жауабы: $x \in(-\infty ; 1) \cup(1 ;+\infty)$

Шешуі:

$${3{x^2} - 2x + 1 \ge 3(2x + 4)}$$ $${3{x^2} - 2x + 1 \ge 6x + 12}$$ $${3{x^2} - 8x - 11 = 0}$$ $${3(x + 1)\left( {x - \frac{{11}}{3}} \right) \ge 0}$$

$${3(x + 1)\left( {x - \frac{{11}}{3}} \right) = 0}$$ $${x = - 1,\quad x = \frac{{11}}{3}}$$

Жауабы: $x \in(-\infty ;-1] \cup\left[\dfrac{11}{3} ;+\infty\right)$

Шешуі:

$${{x^2} + 10x - 5(2x + 5) \le 20 \cdot 10}$$ $${{x^2} + 10x - 10x - 25 - 200 \le 0}$$ $${{x^2} - 225 \le 0}$$ $${(x - 15)(x + 15) \le 0}$$

$${(x - 15)(x + 15) = 0}$$ $${x = 15,\quad x = - 15}$$

Жауабы: $x \in[-15 ; 15]$

Шешуі:

$${ - {x^2} + 14x - 49 \ge 0}$$ $${ - \left( {{x^2} - 14x + 49} \right) \ge 0}$$ $${ - {{(x - 7)}^2} \ge 0}$$

$${ - {{(x - 7)}^2} = 0}$$ $${x = 7}$$

Жауабы: $x=7$

Шешуі:

$${{x^2} - 5x + 16 = 0}$$ $${D = 25 - 64 = - 39 \lt 0}$$ $${a = 1 \gt 0}$$

Жауабы: $x \in R$

Шешуі:

$${4\left( {6{x^2} + 1} \right) \gt 20x - {x^2}}$$ $${24{x^2} + 4 - 20x + {x^2} \gt 0}$$ $${25{x^2} - 20x + 4 \gt 0}$$ $${25{x^2} - 20x + 4 = 0}$$

$${D = 400 - 400 = 0}$$ $${x = \frac{{20}}{{50}} = \frac{2}{5}}$$ $${25{{\left( {x - \frac{2}{5}} \right)}^2} \gt 0}$$

Жауабы: $x \in\left(-\infty ; \dfrac{2}{5}\right) \cup\left(\dfrac{2}{5} ;+\infty\right)$

Шешуі:

$${{{(x + 2)}^2} \le 0}$$ $${{{(x + 2)}^2} = 0}$$

$${x = - 2}$$

Жауабы: $x=-2$

Шешуі:

$${(2 - x)(x - 3)(x - 12) = 0}$$ $${x = 2,\quad x = 3,\quad x = 12}$$

Жауабы: $x \in(-\infty ; 2] \cup[3 ; 12]$

Шешуі:

$${(x + 14)(8 - x)(5 + x) = 0}$$ $${x = - 14,\quad x = 8,\quad x = - 5}$$

Жауабы: $x \in[-14 ;-5] \cup[8 ;+\infty)$

Шешуі:

$${x\left( {{x^2} - 25} \right) \le 0}$$ $${x(x - 5)(x + 5) \le 0}$$

$${x(x - 5)(x + 5) = 0}$$ $${x = 0,\quad x = 5,\quad x = - 5}$$

Жауабы: $x \in(-\infty ;-5] \cup[0 ; 5]$

Шешуі:

$${x(x + 1)(x - 4) \gt 0}$$ $${x(x + 1)(x - 4) = 0}$$

$${x = 0,\quad x = - 1,\quad x = 4}$$

Жауабы: $x \in(-1 ; 0) \cup(4 ;+\infty)$

Шешуі:

$${\left( {{x^2} - 9} \right)\left( {{x^2} - 4} \right) \le 0}$$ $${(x - 3)(x + 3)(x - 2)(x + 2) = 0}$$ $${x = 3,\quad x = - 3,\quad x = 2,\quad x = - 2}$$

Жауабы: $x \in[-3 ;-2] \cup[2 ; 3]$

Шешуі:

$${{x^4} - {x^2} - 4{x^2} + 4 \lt 0}$$ $${{x^2}\left( {{x^2} - 1} \right) - 4\left( {{x^2} - 1} \right) \lt 0}$$ $${\left( {{x^2} - 1} \right)\left( {{x^2} - 4} \right) \lt 0}$$ $${(x - 1)(x + 1)(x - 2)(x + 2) \lt 0}$$ $${(x - 1)(x + 1)(x - 2)(x + 2) = 0}$$ $${x = 1,\quad x = - 1,\quad x = 2,\quad x = - 2}$$

Жауабы: $x \in(-2 ;-1) \cup(1 ; 2)$

Шешуі:

$${{x^2} + 20 \gt 9x}$$ $${{x^2} - 9x + 20 \gt 0}$$ $${(x - 4)(x - 5) \gt 0}$$

$${(x - 4)(x - 5) = 0}$$ $${x = 4,\quad x = 5}$$

Жауабы: $x \in(-\infty ; 4) \cup(5 ;+\infty)$

Шешуі:

$${(2 - x)(3x + 1)(2x - 3) = 0}$$ $${2 - x = 0;\quad 3x + 1 = 0,\quad 2x - 3 = 0}$$ $${x = 2;\quad \quad x = - \frac{1}{3};\quad \quad \quad x = \frac{3}{2}}$$

Жауабы: $x \in\left[-\dfrac{1}{3} ; \dfrac{3}{2}\right] \cup[2 ;+\infty)$

Шешуі:

$${{x^2}(x - 3) - (x - 3) \gt 0}$$ $${(x - 3)\left( {{x^2} - 1} \right) \gt 0}$$ $${(x - 3)(x - 1)(x + 1) \gt 0}$$

$${(x - 3)(x - 1)(x + 1) = 0}$$ $${x = 3,\quad x = 1,\quad x = - 1}$$

Жауабы: $x \in(-1 ; 1) \cup(3 ;+\infty)$

Шешуі:

$${(x - 2)(x - 3)\left( {{x^2} - 1} \right) \ge 0}$$ $${(x - 2)(x - 3)(x - 1)(x + 1) \ge 0}$$ $${(x - 2)(x - 3)(x - 1)(x + 1) = 0}$$ $${x = 2,\quad x = 3,\quad x = 1,\quad x = - 1}$$

Жауабы: $x \in(-\infty ;-1] \cup[1 ; 2] \cup[3 ;+\infty)$

Шешуі:

$${ - \left( {7{x^2} + 6x - 1} \right)(x - 5) \ge 0}$$ $${ - 7(x + 1)\left( {x - \frac{1}{7}} \right)(x - 5) \ge 0}$$

$${ - 7(x + 1)\left( {x - \frac{1}{7}} \right)(x - 5) = 0}$$ $${x = - 1,\quad x = \frac{1}{7},\quad x = 5}$$

Жауабы: $x \in(-\infty ;-1] \cup\left[\dfrac{1}{7} ; 5\right]$

Шешуі:

$${ - x(x + 1)\left( {{x^2} - 49} \right) \lt 0}$$ $${ - x(x + 1)(x - 7)(x + 7) \lt 0}$$ $${ - x(x + 1)(x - 7)(x + 7) = 0}$$ $${x = 0,\quad x = - 1,\quad x = 7,\quad x = - 7}$$

Жауабы: $x \in(-\infty ;-7) \cup(-1 ; 0) \cup(7 ;+\infty)$

Шешуі:

$${{x^2}(x + 6) - (x + 6) \lt 0}$$ $${(x + 6)\left( {{x^2} - 1} \right) \lt 0}$$ $${(x + 6)(x - 1)(x + 1) \lt 0}$$

$${(x + 6)(x - 1)(x + 1) = 0}$$ $${x = - 6,\quad x = 1,\quad x = - 1}$$

Жауабы: $x \in(-\infty ;-6) \cup(-1 ; 1)$

Шешуі:

$${{x^4} - {x^2} - 9{x^2} + 9 \le 0}$$ $${{x^2}\left( {{x^2} - 1} \right) - 9\left( {{x^2} - 1} \right) \le 0}$$ $${\left( {{x^2} - 1} \right)\left( {{x^2} - 9} \right) \le 0}$$

$${(x - 1)(x + 1)(x - 3)(x + 3) \le 0}$$ $${(x - 1)(x + 1)(x - 3)(x + 3) = 0}$$ $${x = 1,\quad x = - 1,\quad x = 3,\quad x = - 3}$$

Жауабы: $x \in[-3 ;-1] \cup[1 ; 3]$

Шешуі:

$${{x^2} + 1 - 3x + {x^2} - 3 \lt 0}$$ $${2{x^2} - 3x - 2 \lt 0}$$ $${2\left( {x + \frac{1}{2}} \right)(x - 2) \lt 0}$$

$${2\left( {x + \frac{1}{2}} \right)(x - 2) = 0}$$ $${x = - \frac{1}{2},\quad x = 2}$$

Жауабы: $x \in\left(-\dfrac{1}{2} ; 2\right)$

Шешуі:

$${(x + 4)(x + 1)(3 - x) \lt 0}$$ $${x + 4 = 0,\quad x + 1 = 0,\quad 3 - x = 0}$$ $${x = - 4,\quad \quad x = - 1,\quad \quad x = 3}$$

Жауабы: $x \in(-4 ;-1) \cup(3 ;+\infty)$

Шешуі:

$${x\left( {64{x^2} - 1} \right) \ge 0}$$ $${x(8x - 1)(8x + 1) \ge 0}$$

$${x = 0,\quad 8x - 1 = 0,\quad 8x + 1 = 0}$$ $${\quad \quad \quad \quad x = \frac{1}{8}\quad \quad \quad x = - \frac{1}{8}}$$

Жауабы: $x \in\left[-\dfrac{1}{8} ; 0\right] \cup\left[\dfrac{1}{8} ;+\infty\right)$

Шешуі:

$${(10 - x)(x + 11)(x + 3) \le 0}$$ $${10 - x = 0,\quad x + 11 = 0,\quad x + 3 = 0}$$ $${x = 10,\quad \quad x = - 11,\quad \quad x = - 3}$$

Жауабы: $x \in[-11 ;-3] \cup[10 ;+\infty)$