Сызықтық теңсіздіктер (Рустюмова 1.5.1)

Шешуі: $$ — 2x + 5x \lt 12 — 3$$ $$3x \lt 9$$ $$x \lt 3$$ Жауабы: 2

Шешуі: $$x — 8 \ge 2x + 1 + 7$$ $$x — 2x \ge 1 + 7 + 8$$ $$ — x \ge 16$$ $$x \le — 16$$ Жауабы: -16

Шешуі: $${x^2} + 3x \gt {x^2} + 3x + x + 3$$ $$ — x \gt 3$$ $$x \lt — 3$$ Жауабы: -4

Шешуі: $$9(9x + 2) — 10(10x — 2) \gt 2 \cdot 90$$ $$81x + 18 — 100x + 20 \gt 180$$ $$ — 19x \gt 180 — 20 — 18$$ $$ — 19x \gt 142$$ $$x \lt — \frac{{142}}{{19}}$$ $$x \lt — 7\frac{9}{{19}}$$ Жауабы: -8

Шешуі: $$14(3x — 1) — 35(x + 1) \lt 70 — 10x$$ $$42x — 14 — 35x — 35 \lt 70 — 10x$$ $$42x — 35x + 10x \lt 70 + 14 + 35$$ $$17x \lt 119$$ $$x \lt 7$$ Жауабы: 6

Шешуі: $$3x + 3 \lt 5x + 5 — 2$$ $$3x — 5x \lt 5 — 2 — 3$$ $$ — 2x \lt 0$$ $$x \gt 0$$ Жауабы: 1

Шешуі: $$3(3x + 5) — 12 \le 4(x — 2) + 12x$$ $$9x + 15 — 12 \le 4x — 8 + 12x$$ $$9x — 4x — 12x \le — 8 — 15 + 12$$ $$ — 7x \le — 11$$ $$x \ge \frac{{11}}{7}$$ $$x \ge 1\frac{4}{7}$$ Жауабы: 2

Шешуі: $$2(x — 3) — 1 \gt 3(x — 2) — 4(x + 1)$$ $$2x — 7 \gt — x — 10$$ $$2x + x \gt — 10 + 7$$ $$3x \gt — 3$$ $$x \gt — 1$$ Жауабы: 0

Шешуі: $${x^2} + x \lt {x^2} + 5x + 5$$ $$x — 5x \lt 5$$ $$ — 4x \lt 5$$ $$x \gt — \frac{5}{4}$$ $$x \gt — 1\frac{1}{4}$$ Жауабы: -1

Шешуі: $$4(x + 4) — 7(x + 7) \lt — 3 \cdot 28$$ $$4x + 16 — 7x — 49 \lt — 84$$ $$4x — 7x \lt — 84 — 16 + 49$$ $$ — 3x \lt — 51$$ $$x \gt 17$$ Жауабы: 18

Шешуі: $$2 + 5 \lt 3x \lt 4 + 5$$ $$7 \lt 3x \lt 9$$ $$\frac{7}{3} \lt x \lt 3$$ $$2\frac{1}{3} \lt x \lt 3$$ Жауабы: $\emptyset$

Шешуі: $$ — 2 — 4 \le — 2x \le 2 — 4$$ $$ — 6 \le — 2x \le — 2$$ $$3 \ge x \ge 1$$ $$1 \le x \le 3$$ Жауабы: 1

Шешуі: $$\left\{ \begin{array}{l}3 — x \gt x\\3 — x \le 11\end{array} \right. \Rightarrow \left\{ \begin{array}{l} — x — x \gt — 3\\ — x \le 11 — 3\end{array} \right. \Rightarrow $$ $$ \Rightarrow \left\{ \begin{array}{l} — 2x \gt — 3\\ — x \le 8\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \lt \frac{3}{2}\\x \ge — 8\end{array} \right.$$ Жауабы: -8

Шешуі: $$\left\{ \begin{array}{l}12{x^2} — 12{x^2} — 2x + 18x + 3 \gt x\\25{x^2} — 1 — 25{x^2} \gt x — 6\end{array} \right. \Rightarrow \left\{ \begin{array}{l} — 2x + 18x — x \gt — 3\\ — x \gt — 6 + 1\end{array} \right. \Rightarrow $$ $$ \Rightarrow \left\{ \begin{array}{l}15x \gt — 3\\ — x \gt — 5\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \gt — \frac{1}{5}\\x \lt 5\end{array} \right.$$ $$ x\in (-\frac{1}{5};\, 5) $$ Жауабы: 0

Шешуі: $$\left\{ {\begin{array}{*{20}{l}}{3x — 8x \lt 6 + 4}\\{2x — 5x \gt — 4 + 1}\\{11x — 15x \lt 3 + 9}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}{ — 5x \lt 10}\\{ — 3x \gt — 3}\\{ — 4x \lt 12}\end{array} \Rightarrow \left\{ {\begin{array}{*{20}{l}}{x \gt — 2}\\{x \lt 1}\\{x \gt — 3}\end{array}} \right.} \right.$$ $$x \in \left( { — 2;1} \right)$$ Жауабы: -1

Шешуі: $$\left\{ {\begin{array}{*{20}{l}}{5(7 — x) — 30 \le 2(3 + 4x) — 40}\\{5x + 15(4 — x) \gt 6(4 — x)}\end{array}} \right. \Rightarrow \left\{ \begin{array}{l}35 — 5x — 30 \le 6 + 8x — 40\\5x + 60 — 15x \gt 24 — 6x\end{array} \right. \Rightarrow $$ $$ \Rightarrow \left\{ \begin{array}{l} — 13x \le — 39\\ — 4x \gt — 36\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \ge 3\\x \lt 9\end{array} \right.$$ $$x \in \left[ {3;9} \right)$$ Жауабы: 3

Шешуі: $${\begin{array}{*{20}{l}}{1,2x + 7 \lt 2x — 3,6}\\{5x — 9x \gt — 63 — 17}\end{array} \Rightarrow \left\{ {\begin{array}{*{20}{l}}{ — 0,8x \lt — 10,6}\\{ — 4x \gt — 80}\end{array} \Rightarrow \left\{ {\begin{array}{*{20}{l}}{x \gt 13,25}\\{x \lt 20}\end{array}} \right.} \right.}$$ $$x \in \left( {13,25;\,\,20} \right)$$ Жауабы: 14

Шешуі: $$\left\{ \begin{array}{l}33x \gt 22 — 2x + 13\\3x + 12(x — 7) \lt 2(3x — 20)\end{array} \right. \Rightarrow \left\{ \begin{array}{l}33x + 2x \gt 22 + 13\\3x + 12x — 84 \lt 6x — 40\end{array} \right. \Rightarrow $$ $$\left\{ \begin{array}{l}35x \gt 35\\3x + 12x — 6x \lt — 40 + 84\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \gt 1\\9x \lt 44\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \gt 1\\x \lt 4\frac{8}{9}\end{array} \right.$$ $$x \in \left( {1;4\frac{1}{9}} \right)$$ Жауабы: 2+3+4=9

Шешуі: $$\left\{ {\begin{array}{*{20}{l}}{6(x — 1) — 4(x — 2) \ge 3(x — 3) — 12x}\\{ — 0,5x — x \gt — 4 — 1}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}{6x — 6 — 4x + 8 \ge 3x — 9 — 12x}\\{ — 1,5x \gt — 5}\end{array}} \right. \Rightarrow $$ $$ \Rightarrow \left\{ {\begin{array}{*{20}{l}}{11x \ge — 11}\\{x \lt \frac{{10}}{3}}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}{x \ge — 1}\\{x \lt \frac{{10}}{3}}\end{array}} \right.$$ $$x \in \left[ {1;\frac{{10}}{3}} \right)$$ Жауабы: -1+1+2+3=5

Шешуі: $$\left\{ \begin{array}{l}12x — 6(x — 1) — 4(x + 2) \le 3(x — 3)\\1,5x — x \lt 5,05\end{array} \right. \Rightarrow $$ $$ \Rightarrow \left\{ \begin{array}{l}12x — 6x + 6 — 4x — 8 \le 3x — 9\\0,5x \lt 5,05\end{array} \right. \Rightarrow \left\{ \begin{array}{l} — x \le — 7\\x \lt 10,5\end{array} \right. \Rightarrow \left\{ \begin{array}{l}x \ge 7\\x \lt 10,5\end{array} \right.$$ $$x \in \left[ {7;\,\,10,5} \right)$$ Жауабы: 7+8+9+10=34

Шешуі: $$\left\{ {\begin{array}{*{20}{l}}{3 — 2x \le 5(x — 2) + 3x}\\{1 — 3x \ge 4(5x — 1) — 21x}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}{3 — 2x \le 5x — 10 + 3x}\\{1 — 3x \ge 20x — 4 — 21x}\end{array}} \right. \Rightarrow $$ $$ \Rightarrow \left\{ {\begin{array}{*{20}{l}}{ — 2x — 5x — 3x \le — 10 — 3}\\{ — 3x — 20x + 21x \ge — 4 — 1}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}{ — 10x \le — 13}\\{ — 2x \ge — 5}\end{array}} \right. \Rightarrow \left\{ {\begin{array}{*{20}{l}}{x \ge 1,3}\\{x \le 2,5}\end{array}} \right.$$ $$x \in \left[ {1,3;\,\,2,5} \right]$$ Жауабы: 2