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