Алгебра 10-11 класс. Системы неравенств
| Задача 1. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{2^x} + 17 \cdot {2^{3 — x}} \leqslant 25\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {\dfrac{{{x^2} — 3x — 5}}{{x — 4}} + \dfrac{{3{x^2} — 15x + 2}}{{x — 5}} \leqslant 4x + 1} \end{array}} \right.\)
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| Задача 2. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{4^x} — 29 \cdot {2^x} + 168 \leqslant 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {\dfrac{{{x^4} — 5{x^3} + 3x — 25}}{{{x^2} — 5x}} \geqslant {x^2} — \dfrac{1}{{x — 4}} + \dfrac{5}{x}} \end{array}} \right.\)
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| Задача 3. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {\dfrac{{{2^{4x + 2}}}}{{{4^{x + 1}}}} > 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {1 + {{\log }_3}\left( {x — 4} \right) \leqslant {{\log }_3}\left( {x + 21} \right)} \end{array}} \right.\)
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| Задача 4. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{9^x} — 10 \cdot {3^x} + 9 < 0} \\ {\dfrac{2}{x} < 2 + \dfrac{3}{{x — 1}}\,\,\,\,\,\,\,\,\,\,\,} \end{array}} \right.\)
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| Задача 5. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{3^{x + 2}} + {9^{x + 1}} — 810 > 0\,\,\,\,\,\,\,} \\ {\log _3^2x + 4{{\log }_3}x + 3 \geqslant 0} \end{array}} \right.\)
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| Задача 6. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{9^{x + 1}} + 3 \geqslant 28 \cdot {3^x}\,\,\,} \\ {{{\log }_2}\left( {{x^2} — 2x} \right) \leqslant 3} \end{array}} \right.\)
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| Задача 7. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{{\log }_2}\dfrac{{3x — 2}}{{x — 1}} + 3{{\log }_8}\dfrac{{{{\left( {x — 1} \right)}^3}}}{{3x — 2}}\, < 1} \\ {\dfrac{{\sqrt {8 — 2x — {x^2}} }}{{2x + 9}} \geqslant \dfrac{{\sqrt {8 — 2x — {x^2}} }}{{x + 10}}\,\,\,} \end{array}} \right.\)
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| Задача 8. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {\dfrac{{3 \cdot {{64}^x} + {2^x} — 70}}{{{{64}^x} — 2}} \geqslant 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {\log _3^2\left( {x + 3} \right) — 3{{\log }_3}\left( {x + 3} \right) + 2 \leqslant 0} \end{array}} \right.\)
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| Задача 9. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{2^{{x^2} + 3x — 3}} — {2^{{x^2} + 3x — 5}} — 96 \leqslant 0} \\ {{{\log }_{\dfrac{1}{3}}}\,\dfrac{{2x — 1}}{{x + 2}} > 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \end{array}} \right.\)
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| Задача 10. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {2 \cdot {3^{2x + 4}} — 245 \cdot {3^x} + 3 \leqslant 0} \\ {{{\log }_2}\left( {{x^2} + 4x + 5} \right) > 2\,\,\,\,\,\,\,\,} \end{array}} \right.\)
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| Задача 11. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {\dfrac{{{7^x} — 30}}{{{7^{x — 1}} + 1}} \leqslant — 14\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {{{\log }_3}\left( {1 — 2x} \right) \geqslant {{\log }_3}\left( {5x — 2} \right)} \end{array}} \right.\)
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| Задача 12. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {4 \cdot {3^{x + 2}} — 2 \cdot {5^{x + 2}} \leqslant {5^{x + 3}} — {3^{x + 3}}} \\ {\lg \left( {{x^2} — 2x — 2} \right) \leqslant 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \end{array}} \right.\)
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| Задача 13. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{5^{2x + 1}} > {5^x} + 4\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {{{\log }_3}\left( {{x^2} — x} \right) \geqslant {{\log }_3}\left( {3x + 2} \right)} \end{array}} \right.\)
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| Задача 14. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{3^x}\, < 1 + 12 \cdot {3^{ — x}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {2\,\,\ln \dfrac{1}{{3x — 2}} + \ln \left( {5 — 2x} \right) \geqslant 0} \end{array}} \right.\)
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| Задача 15. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {\dfrac{{x — 2\sqrt x — 8}}{{{2^x} — 4}} \geqslant 0\,\,\,\,\,\,\,\,\,\,\,} \\ {\dfrac{{{{\log }_2}x — 5}}{{1 — 2{{\log }_x}2}} \geqslant 2{{\log }_2}x} \end{array}} \right.\)
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| Задача 16. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{{\left( {\dfrac{1}{2}} \right)}^{{{\log }_{\frac{1}{9}}}\left( {2{x^2} — 3x + 1} \right)}}\, < 1\,\,\,\,\,\,\,\,\,} \\ {2 + \dfrac{{\log _2^2x}}{{1 + {{\log }_2}x}} > {{\log }_2}x} \end{array}} \right.\)
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| Задача 17. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {\dfrac{{{4^x}}}{{{2^x} — 1}} \leqslant \dfrac{{{2^x} + 12}}{3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {{{\log }_4}\left( {3 \cdot {4^{x + 1}} — 8} \right) < 2x + 1} \end{array}} \right.\)
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| Задача 18. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{4^x} \leqslant 9 \cdot {2^x} + 22\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {{{\log }_3}\left( {{x^2} — x — 2} \right) \leqslant 1 + {{\log }_3}\dfrac{{x + 1}}{{x — 2}}} \end{array}} \right.\)
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| Задача 19. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{{\log }_{6 — x}}\dfrac{{{{\left( {x — 6} \right)}^2}}}{{x — 2}} \geqslant 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {\dfrac{{{x^2} — x — 14}}{{x — 4}} + \dfrac{{{x^2} — 8x + 3}}{{x — 8}} \leqslant 2x + 3} \end{array}} \right.\)
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| Задача 20. Решите систему неравенств \(\left\{ {\begin{array}{*{20}{c}} {{{\log }_{7 — 2x}}\left( {x + 6} \right) \leqslant 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\ {x — \dfrac{{x — 3}}{{x + 6}} — \dfrac{{{x^2} + 27x + 90}}{{{x^2} + 8x + 12}} \leqslant — 1} \end{array}} \right.\)
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