>Логарифмічні рівняння>Як розв'язувати показникові нерівності

Як розв'язувати показникові нерівності

Пiд час розв’язування показникових нерiвностей важливо звертати увагу на значення аргументу $a$ в логарифмi. Це значення визначає, чи потрiбно перевертати знак нерiвностi пiд час множення чи дiлення на натуральний логарифм $\ln a$ або на десятковий логарифм $\log a$ .

Правило

Розв’язування показникових нерiвностей

Якщо

a > 1

\ln (a) > 0

i нерiвнiсть можна розв’язати як звичайно.

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & > \frac{\ln b}{\ln a} & x & > \frac{\log b}{\log a} \end{array}

Якщо $0 < a < 1$ , $\ln (a) < 0$ , доведеться перевернути знак нерiвностi, коли ми в кiнцевому пiдсумку дiлитимемо або множитимемо на вiд’ємне число!

\begin{array}{l} a^{x} & > b & a^{x} & > b \\ \ln a^{x} & > \ln b & \log a^{x} & > \log b \\ x \ln a & > \ln b & x \log a & > \log b \\ x & < \frac{\ln b}{\ln a} & x & < \frac{\log b}{\log a} \end{array}

Приклад 1

Розв’яжи нерiвнiсть ${3.5}^{x} > 439$

\begin{array}{l} 3.5 x > 439 \ln {3.5}^{x} & > \ln 439 x \ln 3.5 & > \ln 439 & | : \ln 3.5 x & > \frac{\ln 439}{\ln 3.5} \approx 4.9Приклад 2 Розв’яжи нерiвнiсть 50 \cdot {1.05}^{x} > 300 \begin{array}{l} 50 \cdot {1.05}^{x} > 300 | : 50 {1.05}^{x} & > 6 \ln {1.05}^{x} & > \ln 6 x \ln 1.05 & > \ln 6 & | : \ln 1.05 x & > \frac{\ln 6}{\ln 1.05} \approx 36.7Приклад 3 Розв’яжи нерiвнiсть 3 \cdot {0.25}^{x} > 27 \begin{array}{l} 3 \cdot {0.25}^{x} > 27 | : 3 {0.25}^{x} & > 9 \lg {0.25}^{x} & > \lg 9 x \lg 0.25 & > \lg 9 & | : \log 0.25 x & < \frac{\lg 9}{\log 0.25} \approx - 1.6Попередня стаття Що таке логарифмічні рівняння? Наступна стаття Як розв'язувати логарифмічні нерівностіПовернутисяДодому Навчальний план Статистика ЛігаПро нас Про House of Math Співробітники Працевлаштування в House of Math Медіа Лекції Блог Контактна інформація support@houseofmath.com Адреса House of Math 470 Ramona Street Palo Alto, CA 94301 USA FAQ Корисні посилання WeTakeAction Магія математики для України Репетиторство Джуніор-матем Вправи з ручкою та папером Енциклопедія Мова/Навчальна програма Умови Політика конфіденційності Політика щодо файлів cookies Завантажте наш додаток{"props":{"pageProps":{"campaign":false,"articleBody":" Пiд час розв’язування показникових нерiвностей важливо звертати увагу на значення аргументу a в логарифмi. Це значення визначає, чи потрiбно перевертати знак нерiвностi пiд час множення чи дiлення на натуральний логарифм ln ⁡ a або на десятковий логарифм log ⁡ a. Правило Розв’язування показникових нерiвностей Якщо a \u003e 1, ln ⁡ (a) \u003e 0 i нерiвнiсть можна розв’язати як звичайно. ax \u003e b ax \u003e b ln ⁡ ax \u003e ln ⁡ b log ⁡ ax \u003e log ⁡ b x ln ⁡ a \u003e ln ⁡ b x log ⁡ a \u003e log ⁡ b x \u003e ln ⁡ b ln ⁡ a x \u003e log ⁡ b log ⁡ a Якщо 0 \u003c a \u003c 1, ln ⁡ (a) \u003c 0, доведеться перевернути знак нерiвностi, коли ми в кiнцевому пiдсумку дiлитимемо або множитимемо на вiд’ємне число! ax \u003e b ax \u003e b ln ⁡ ax \u003e ln ⁡ b log ⁡ ax \u003e log ⁡ b x ln ⁡ a \u003e ln ⁡ b x log ⁡ a \u003e log ⁡ b x \u003c ln ⁡ b ln ⁡ a x \u003c log ⁡ b log ⁡ a Приклад 1 Розв’яжи нерiвнiсть 3.5x \u003e 439 3.5x \u003e 439 ln ⁡ 3.5x \u003e ln ⁡ 439 x ln ⁡ 3.5 \u003e ln ⁡ 439 |: ln ⁡ 3.5 x \u003e ln ⁡ 439 ln ⁡ 3.5 ≈4.9 Приклад 2 Розв’яжи нерiвнiсть 50 ⋅1.05x \u003e 300 50 ⋅1.05x \u003e 300 |:50 1.05x \u003e 6 ln ⁡ 1.05x \u003e ln ⁡ 6 x ln ⁡ 1.05 \u003e ln ⁡ 6 |: ln ⁡ 1.05 x \u003e ln ⁡ 6 ln ⁡ 1.05 ≈36.7 Приклад 3 Розв’яжи нерiвнiсть 3 ⋅0.25x \u003e 27 3 ⋅0.25x \u003e 27 |:3 0.25x \u003e 9 lg ⁡ 0.25x \u003e lg ⁡ 9 xlg ⁡ 0.25 \u003e lg ⁡ 9 |: log ⁡ 0.25 x \u003c lg ⁡ 9 log ⁡ 0.25 ≈−1.6 ","topic":{"breadcrumbs":[{"_key":"c0cf1a4b-5f7d-5390-9a52-e6ae2428b50c","h1":"Енциклопедія","path":"/encyclopedia"},{"_key":"58d8130f-e1b9-56b5-a4aa-04a4be9d7874","h1":"Алгебра","path":"/encyclopedia/alhebra"},{"_key":"c06d9fed-d452-510c-9284-c1dc6294a8c5","h1":"Логарифми","path":"/encyclopedia/alhebra/loharyfmy"},{"_key":"f05334ae-9371-5feb-aa08-b192c5696de2","h1":"Логарифмічні рівняння","path":"/encyclopedia/alhebra/loharyfmy/loharyfmichni-rivnyannya"},{"_key":"f396fd80-c50d-5f46-bd48-eb35c5657935","h1":"Як розв'язувати показникові нерівності","path":"/encyclopedia/alhebra/loharyfmy/loharyfmichni-rivnyannya/yak-rozvyazuvaty-pokaznykovi-nerivnosti"}],"cssUrl":"/8c2ce5fa-9e80-4618-888c-fb8e70afc5d1-uk-5.css","h1":"Як розв'язувати показникові нерівності","htmlContent":{"html":" \u003c/p\u003e\u003cp class=\"indent\" \u003e     Пiд час розв’язування показникових нерiвностей важливо звертати увагу на значення аргументу \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e в логарифмi. Це значення визначає, чи потрiбно перевертати знак нерiвностi пiд час множення чи дiлення на натуральний логарифм \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math\u003e або на десятковий логарифм \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/math type=\"punct\"\u003e. \u003c/p\u003e      \u003cdiv class=\"tcolorbox foorule\" id=\"tcolobox-620\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                        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class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmn\u003e0\u003c/mn\u003e\u003c/math\u003e, доведеться перевернути знак нерiвностi, коли ми в кiнцевому пiдсумку дiлитимемо або множитимемо на вiд’ємне число! \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"align-star\"\u003e                        \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                       \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmsup\u003e\u003cmrow  \u003e\u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e                    \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                       \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003C;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003eb\u003c/mi\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e log\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmi  \u003ea\u003c/mi\u003e\u003c/mrow\u003e\u003c/mfrac\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                   \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e                                                                                                                                                                                                                        \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-621\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Приклад 1\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003eРозв’яжи нерiвнiсть \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                           \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                    \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                     \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-label\"\u003e                           \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-label\"\u003e                           \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                  \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                          \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003cmo  class=\"MathClass-punc\" type=\"divide\"\u003e:\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e                                  \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e4\u003c/mn\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mrow\u003e   \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e  \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e4\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e9\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                            \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                                 \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                                          \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-622\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Приклад 2\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003eРозв’яжи нерiвнiсть \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                         \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                            \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003cmo  class=\"MathClass-punc\" type=\"divide\"\u003e:\u003c/mo\u003e\u003cmn\u003e5\u003c/mn\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                             \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                        \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-label\"\u003e                         \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                  \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-label\"\u003e                         \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                   \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-label\"\u003e                         \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                    \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                        \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003cmo  class=\"MathClass-punc\" type=\"divide\"\u003e:\u003c/mo\u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e                               \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mtd\u003e                           \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e  \u003cmfrac\u003e\u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmn\u003e6\u003c/mn\u003e\u003c/mrow\u003e  \u003cmrow  \u003e\u003cmi class=\"qopname\"\u003e ln\u003c/mi\u003e\u003cmo\u003e ⁡ \u003c/mo\u003e \u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e1\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003c/mfrac\u003e \u003cmo  class=\"MathClass-rel\"\u003e≈\u003c/mo\u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmn\u003e6\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                              \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                              \u003cmtd  class=\"align-label\"\u003e \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003c/mtable\u003e\u003c/math\u003e   \u003c/div\u003e  \u003c/div\u003e \u003cp class=\"noindent\" \u003e\u003c/b\u003e                                                                                                                                                                                                                    \u003c/p\u003e      \u003cdiv class=\"tcolorbox fooqaexample\" id=\"tcolobox-623\"\u003e     \u003cdiv class=\"tcolorbox-title\"\u003e                                                                                                                                                                                                                                          \u003cp class=\"indent\" \u003e     Приклад 3\u003c/p\u003e\u003c/div\u003e  \u003cdiv class=\"tcolorbox-content\"\u003e\u003cp class=\"noindent\" \u003e\u003cb\u003eРозв’яжи нерiвнiсть \u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"inline\" \u003e\u003cmstyle mathvariant=\"bold-italic\"\u003e\u003cmstyle mathvariant=\"bold\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003c/mstyle\u003e\u003c/mstyle\u003e\u003c/math\u003e \u003c/p\u003e\u003cmath mathvariant=\"normal\"   xmlns=\"http://www.w3.org/1998/Math/MathML\"   display=\"block\" \u003e\u003cmtable  columnalign=\"left\" class=\"alignat-star\"\u003e                           \u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmn\u003e3\u003c/mn\u003e \u003cmo  class=\"MathClass-bin\"\u003e⋅\u003c/mo\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                     \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e7\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                 \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e\u003c/mrow\u003e\u003cmo class=\"MathClass-close\" fence=\"true\" mathsize=\"1.19em\" \u003e|\u003c/mo\u003e\u003c/mrow\u003e\u003cmo  class=\"MathClass-punc\" type=\"divide\"\u003e:\u003c/mo\u003e\u003cmn\u003e3\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                       \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                               \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e\u003c/mtr\u003e\u003cmtr\u003e\u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003cmsup\u003e\u003cmrow  \u003e\u003cmstyle  class=\"text\"\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003cmstyle  class=\"math\"\u003e\u003cmn\u003e0\u003c/mn\u003e\u003cmo  class=\"MathClass-punc\" type=\"punct\"\u003e.\u003c/mo\u003e\u003cmn\u003e2\u003c/mn\u003e\u003cmn\u003e5\u003c/mn\u003e\u003c/mstyle\u003e\u003cmtext  \u003e\u003c/mtext\u003e\u003c/mstyle\u003e\u003c/mrow\u003e\u003cmrow  \u003e\u003cmi  \u003ex\u003c/mi\u003e\u003c/mrow\u003e\u003c/msup  \u003e\u003c/mtd\u003e                                         \u003cmtd  class=\"align-even\"\u003e \u003cmo  class=\"MathClass-rel\"\u003e\u0026#x003E;\u003c/mo\u003e \u003cmn\u003e9\u003c/mn\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                                  \u003cmtd  columnalign=\"right\" class=\"align-odd\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-even\"\u003e\u003cmspace width=\"2em\"/\u003e\u003c/mtd\u003e                                           \u003cmtd  columnalign=\"right\" class=\"align-label\"\u003e\u003c/mtd\u003e                               \u003cmtd  class=\"align-label\"\u003e                                \u003cmspace 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Вивчай математику швидко та легко. 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Повертайся сюди, щоб переглянути тести та оновлення своєї математичної подорожі. Там ти знайдеш такі речі:","chestTitle":"Ласкаво просимо!","close":"Закрити","collection":"Моя колекція","collectionList":{"allItems":"Усі кубки","allRewards":"Усі нагороди","bootcamps":"Bootcamps","competition":"Змагання","mathdrill":"Основи математики","mathEssentials":"Основи математики"},"collectionShowcase":{"bootcamp":"Bootcamp","gathered":"Зібрано","championship-2021":"Чемпіонат з множення 2021"},"rankPopupTitle":"Нова система підвищення рівня XP!","rankPopupText":"Ти можеш здобути XP, виконуючи завдання на порталі. Досягнувши порогу XP, ти підвищиш рівень.","completed":"Завершено","weExchange":"Обмінюємо","dummyData":{"mult":{"title":"Множення","description":"Що таке множення?"},"algebra":{"title":"Алгебра","description":"Елементарна алгебра"}},"event":{"checkQuest":"Прочитано та зрозуміло","INTRO":{"dialogue":[{"text":"Це твоя база. Тут ти отримуватимеш повідомлення та завдання, збираючи скарби та нагороди під час своєї подорожі світом математики.","title":"Ласкаво просимо додому"}]}},"findYourLearningPath":"Знайди свій шлях навчання, створи аватар та отримай домашнього дракона!","gemBlueAlt":"Блискучий синьо-білий діамант","gemGreenAlt":"Блискучий зелений смарагд","gemRedAlt":"Сяйнистий червоний рубін","gotoavatarpage":"Перейти до аватара","Home":"Домашня база","homebasepage":"Початкова база:","imageAlt":{"goldenHex":"Золотий шестикутник","levelBadge":"Значок переходу на вищий рівень"},"introText":"Ти саме той герой/героїня, якого ми шукаємо. Ми повернемося до цього. Дозволь спочатку показати тобі все навколо.","introTitle":"Гарний початок!","kudos":"Kudos","level":"РІВЕНЬ","levelUp":"Підвищення рівня","navbarIcon":"Схованка","next":"Далі","shadow":"Тінь","playerName":"Ім'я гравця...","questCompleted":" Тест виконано!","questDescriptions":{"bootcamp":" Досліди bootcamp \u003cb\u003e{{BootcampTitle}}\u003c/b\u003e і здобудь перемогу над босом буткемпів!","essential":"Щоб підтвердити свої навички, пройди Основи математики!","encyclopedia":"Прочитай і зрозумій \u003cb\u003e{{EncyclopediaArticle}}\u003c/b\u003e  та постав відповідну позначку!"},"quests":"Мій математичний тест","mathQuests":"Математичні тести","completedMathQuests":"Виконані завдання","completeQuestsToSee":"Щоб побачити тести тут, потрібно їх виконати","questText":"Пройди діагностичний тест й отримай індивідальний шлях навчання","quiz":{"advancedQuizText":"Щоб рухатися вперед, тепер тобі потрібно лише обрати теми, які тобі знайомі.","advancedQuizTitle":"Чудові результати тесту!","advancedSubjects":{"quiz-subject-advanced-derivatives":{"title":"Похідна"},"quiz-subject-advanced-differential-equations":{"title":"Диференціальні рівняння"},"quiz-subject-advanced-integrals":{"title":"Інтеграли"},"quiz-subject-advanced-logarithms":{"title":"Логарифми"},"quiz-subject-advanced-pi":{"title":"Число «Пі», π"},"quiz-subject-advanced-probability":{"title":"Теорій ймовірностей"},"quiz-subject-advanced-proofs":{"title":"Доведення"},"quiz-subject-advanced-quadratic-equations":{"title":"Квадратні рівняння"},"quiz-subject-advanced-series":{"title":"Числові ряди"},"quiz-subject-advanced-sets-of-numbers":{"title":"Числові множини"},"quiz-subject-advanced-trigonometry":{"title":"Тригонометрія"},"quiz-subject-advanced-vectors":{"title":"Вектори"}},"applyText":"Ти, очевидно, добре знаєш математику! Якщо ти бажаєш доєднатися до команди House of Math, будь ласка, подай заявку за посиланням \u003clinkToCareers\u003ehere\u003c/linkToCareers\u003e.","awardTitle":"Ми перетворюємо твій досвід навчання на гру","awardTitleSubtext":"Під час мандрівки ти знайдеш такі скрині з нагородами.","awesome":"Чудово!","buttonTakeQuiz":"Пройти тест","creatingYourLearningPath":"Створення твого шляху навчання...","learningPath":"Створення твого шляху навчання","greatResultsText":"Щоб рухатися вперед, тепер тобі потрібно лише обрати теми, які тобі знайомі.","greatResultsTitle":"Чудові результати тесту!","subjectKnowledgeRating":{"0":"Жахливо","1":"Погано","2":"Нейтрально","3":"Добре","4":"Чудово"},"subjects":{"quiz-subject-algebraic-factorization":{"questionRating":"Наскільки добре ти опанував/опанувала розкладання многочлена на множники?","title":"Розкладання многочлена на множники"},"quiz-subject-basic-algebra":{"questionRating":"Наскільки добре ти опанував/опанувала операції з математичними знаками?","title":"Основи алгебри"},"quiz-subject-equations":{"questionRating":"Наскільки добре ти опанував/опанувала рівняння?","title":"Рівняння"},"quiz-subject-fractions":{"questionRating":"Наскільки добре ти опанував/опанувала дроби?","title":"Дроби"},"quiz-subject-functions":{"questionRating":"Наскільки добре ти опанував/опанувала функції?","title":"Функції"},"quiz-subject-parentheses":{"questionRating":"Наскільки добре ти опанував/опанувала круглі дужки?","title":"Круглі дужки"},"quiz-subject-percent":{"questionRating":"Наскільки добре ти опанував/опанувала відсоток?","title":"Відсоток"},"quiz-subject-powers":{"questionRating":"Наскільки добре ти опанував/опанувала степінь числа?","title":"Степінь числа"},"quiz-subject-the-multiplication-table":{"questionRating":"Наскільки добре ти опанував/опанувала таблицю множення?","title":"Таблиця множення"}},"title":"Щоб навчання було ефективним, пройди тест!","titleSubtext":"Пройди цей тест, щоб дізнатися, з чого почати.","yesNoQuestionTitle":"Чи працював/працювала ти коли-небудь над \u003clowercase\u003e{{ subject }}\u003c/lowercase\u003e?"},"retakeQuiz":"Повторити тест","reward":"Нагорода","storyCards":{"buttonCool":"Класно!","buttonIWill":"Я буду!","buttonJump":"Стрибай","buttonLetUsTry":"Спробуймо!","buttonNice":"Чудово!","buttonOfCourse":"Звісно!","buttonOh":"О!","buttonPeekaBoo":"Піка Бу","buttonRoar":"Рев","buttonThanks":"Дякую!","buttonWow":"Овва!","buttonYes":"Так!","ChoosePetTrick":"Твій дракон готовий навчитися нового трюку! Обери один із трьох нижче!","ChooseYourEgg":"Дивися, хтось залишив тобі подарунок!","CoolDown":"Тобі потрібно зачекати певний час, перш ніж ти знову зможеш віддати математичну енергію! Використай цей час, щоб зібрати більше енергетичних сфер в  Основах математики","CoolDownFirstTime":"Твоє яйце повне, й потрібно трохи зачекати, перш ніж можна буде віддати більше математичної енергії! Використай цей час, щоб зібрати більше математичної енергії зі скринь в Основах математики!»","DoNotHaveEnoughMnO":"Ти більше не маєш енергії лайків! Щоб здобути більше, виконай тести!","DragonFull":"Твій дракон ситий! Зачекай певний час, перш ніж давати йому енергію лайків.","DragonTrickCooldown":"Твій дракон утомився після навчання нового трюку. Перш ніж пробувати знову, дозволь йому трохи відпочити!","DragonTrickNotWilling":"Твій дракон не бажає тренуватися, доки не буде щасливий і нагодований!","DragonTutorial1":"Дбай про здоров’я свого дракона, годуючи його енергією лайків!","DragonTutorial2":"Ходять чутки, що якщо ти деякий час підтримаєш свого Дракона здоровим, він може навчитися нових трюків!","DragonTutorial3":"Дракони також потребують пестощів! Регулярно гладь свого дракона!!","DragonTutorial4":"Молодець! Твій Дракон щасливий! Продовжуй дарувати йому любов щодня!","EggHasBeenChosen":"Чудовий вибір! Пропоную безпечно помістимо яйце у твою базу. Чи зможемо ми змусити його вилупитися?","EggHungry1":"Гм... . Твоє яйце здається голодним!","EggHungry2":"Ось, візьми цю енергію лайків!","EggHungry3":"Спробуй нагодувати яйце енергією лайків, натиснувши на нього!","EggInfoStillHatching":"Твоє яйце зараз вилуплюється! Проте потрібно згодувати йому ще енергії лайків!","EggIsAboutToHatch":"Неймовірно! Тепер ти маєш Дракона на своїй домашній базі","FeedMathEnergyToEggFirstTime":"Спробуй нагодувати яйце енергією лайків, натиснувши на нього!","FirstTimeGettingMathEnergyFromChest":"Круто! Ти знайшов свою першу енергетичну сферу! Віддай її своєму яйцю на домашній базі!!","GiveDragonName":"Введи ім'я дракона або просто натисни OK.","InfoAboutPetTrick":"Твій дракон може вчитися, коли він ситий і задоволений!","MathEnergyTutorial1":"Щоб яйце вилупилося, потрібно нагодувати його достатньою кількістю енергії лайків.","MathEnergyTutorial2":"Щоб здобути більше енергії лайків, виконай тести!","NextHatchingState1":"Овва! Глянь! Твоє яйце тепер має інший вигляд!","NextHatchingState2":"Твоє яйце тріснуло ще трохи! Цікаво, що всередині...","NextHatchingState3":"Ще трохи... Твоє яйце скоро вилупиться!","TimesTableCompetition2021FirstPlace":"Вітаємо з перемогою в Чемпіонаті з множення 2021 року! Отримай унікальний кубок і додай його до своєї колекції.","TimesTableCompetition2021Participant":"Дивовижна робота! Ти заробив/заробила цей приз, беручи участь у Чемпіонаті з множення 2021 року.","TimesTableCompetition2021SecondPlace":"Вітаємо з 2-м місцем у Чемпіонаті з множення 2021 року! Отримай унікальний кубок і додай його до своєї колекції.","TimesTableCompetition2021ThirdPlace":"Вітаємо з 3-м місцем у Чемпіонаті з множення 2021 року! Отримай унікальний кубок і додай його до своєї колекції."},"space":"Пробіл","takeTheQuiz":"Пройди тест, щоб розблокувати свій дім!","tooltip":{"videoCourse":"Bootcamp","encylopedia":"Енциклопедія","mathEssential":"математичні вишколи","description":{"videoCourse":"Дивися відео, виконуй завдання та здобудь перемогу над босом буткемпів","encylopedia":"Прочитай і зрозумій та постав відповідну позначку!","mathEssential":"Щоб підтвердити свої навички, пройди Основи математики!"}},"titles":{"houseOfMathMember":"Член House of Math","mathApprentice":"Математик-практикант","mathemagician":"Матемаг","mathJunior":"Математик-початківець","mathlete":"Мататлет","mathScholar":"Математик-експерт","mathSorcerer":"Математик-чудотворець"},"unlockYourHomebase":"Увійди або зареєструйся, щоб перейти до особистого кабінету, де можна створити свій індивідуальний шлях навчання, відстежувати здобуті бали досвіду (XP) та винагороди, отримати доступ до своєї Схованки й багато іншого!","updateQuests":"Оновити навчальні тести","xp":"XP","open":"ВІДЧИНЕНО","yourCompletedQuests":"Твої виконані квести","yourFirstQuest":"Твій перший квест","yourNextQuest":"Твій наступний квест","yourUpcomingQuests":"Твої майбутні квести","removeTask":"Видалити зі шляху навчання","cancel":"Скасувати","rank":"Ранг","leagueNavButton":"Ліга","menuLabel":"Меню","loading":"Завантаження","loadscreenHint":{"feedDragon":"Підвищ рейтинг свого дракона, годуючи його енергією лайків","earnKudos":"Заробляй енергію лайків, виконуючи математичні завдання","earnGems":"Заробляй коштовності, виконуючи математичні завдання","buyStuff":"Купуй круті речі з коштовностями в крамниці дракона","wardrobe":"Усі куплені речі зберігаються в шафі","moreMath":"Чим більше математичних завдань ти виконаєш, тим більше отримаєш коштовностей та енергії лайків"}},"menu":{"mathTopics":"Уся математика","bootcamp":"Bootcamps","dashboard":"Додому","digitalTools":"Цифрові засоби","drillhefter":"Основи математики","encyclopedia":"Енциклопедія","exams":"Eksamensoppgaver","exitPreview":"Закрити попередній перегляд","games":"Ігри","logOut":"Вийти","login":"Увійти","mathEssentials":"Основи математики","mathVault":"Джуніор-матем","register":"Зареєструватися","settings":"Налаштування","studyTips":"Поради з навчання","toggleNavigationMenu":"Перемкнути меню навігації","tutors":"Tutor-on-Demand","daysStreak":"{{days}} днів з реєстрації","learningTools":"Навчальні матеріали","moreLearningTools":"Більше навчальних матеріалів","penNPaperExercise":"Вправи з ручкою та папером","excelOrGeoGebra":"Рецепти Excel / GeoGebra","numberNQuantities":"Числа та величини","statNProbability":"Статистика та ймовірність","stackNLoad":"Stack n´load","juniorMath":"Джуніор-матем","windSurfer":"Wind Surfer","treasureTrail":"Treasure Trail","multiplicationMaster":"Multiplication Master","misc":"Профіль","showMore":"Показати більше","showAllGames":"Показати всі ігри","gotoGames":"Перейти до ігор","switchProfile":"Перемкнути профіль","createFamily":"Додати профіль","getHomApp":"Завантажте додаток House of Math","tryFree":"Спробуйте безкоштовно","signUpForFree":"Зареєструватися безплатно","referral":"Запросити друзів"},"navbar":{"explore":"Ознайомитися","menu":"Меню","panelAdmin":"Панель адміністратора","panelTutor":"Панель репетитора","panelStudent":"Учнівська панель","search":"Пошук","languages":"мова","norwegian":"Norsk","english":"English","ukrainian":"Українська","settings":"Налаштування","logOut":"Вийти","dashboard":"Додому","chat":"Чат","userProfile":"Профіль","leaderboard":"Список лідерів"},"notFound":{"clickOne":"Клацни людину, яку, на твою думку, варто звільнити","fire":"Звільнити","goDashboard":"Перейти до панелі керування","pageNotFound":"Сторінку не знайдено","pageRemoved":"Сторінку видалено","punishDevs":"Наших розробників потрібно покарати за це жахливе недбальство","employeesOnLunchBreak":"Здається, наші співробітники пішли на обідню перерву. Ми незабаром повернемося!"},"search":{"adaptedView":"Індивідуальний режим перегляду","bootcamp":"Bootcamps","all":"Усе","drill":"Основи математики","encyclopedia":"Енциклопедія","noResult":"Результатів пошуку не знайдено","readMore":"Дізнатися більше","watchVideo":"Дивитися відео","remove":"Видалити","search":"Пошук..","geogebra":"GeoGebra","searchResultAppear":"Тут з’являться ваші результати пошуку","searchTooltip":"Індивідуальний режим перегляду призначений для перегляду статей рівня, який ти обрав/обрала у налаштуваннях. Перемкни, щоб переглядати всі статті.","software":"Цифрові засоби","title":"Результати пошуку","searchPlaceholderText":"Чого тобі хочеться вивчити?"},"dashboard":{"activity":"Набір завдань","bootcamp":{"buttonContinueLearning":"Продовжити навчання","buttonToOverview":"Перейти до огляду","comingSoon":"Незабаром!","edit":"Редагувати","title":"Мої Bootcamps"},"gamification":{"badge":"Значок Bootcamp","badgeBackground":"Фон бейджа","buttonToBase":"Схованка","gemIcon":"Коштовності Значок","completeQuestsGetTopics":"Виконай тести, щоб отримати доступ до більшої кількості тем"},"completeProfile":{"button":"Завершити реєстрацію","description":"Нам потрібно дізнатися про тебе трохи більше, щоб персоналізувати твій досвід"},"continue":"Продовжити","topicOverview":"Огляд теми","tutor":{"assigningTutor":"Будь ласка, зачекай, доки ми знайдемо для тебе репетитора.","bookPrivateTutoring":"Замовити репетиторство","tutorPending":"Очікуй","tutorScheduling":"Коли тобі буде призначене заняття, воно з’явиться тут.","myTutor":"Мій репетитор","myTutors":"Мої репетитори","myStudentPanel":"Моя учнівська панель"},"activities":"Завдання","myActivities":"Мої завдання","tagNew":"новий","news":{"buttonChat":"Чат","buttonPlay":"Грати","buttonShowMe":"Покажи мені","buttonStart":"Розпочати","campAlt":"Зона Bootcamps","content1":"Долучайся до пригод разом із бджілкою Д’Рілло за допомогою оновленого розділу Основи математики. Дізнайся про найважливіші математичні теми, граючи з веселими завданнями!","content2":"Наші репетитори готові тобі допомогти. Скористайся чатом, щоб отримати відповіді на всі свої запитання з математики.","content3":"Візьми участь у Чемпіонаті з множення, змагайся з іншими та вигравай першокласні призи!","content4":"Bootcamps — це веселі, інтерактивні, анімовані уроки з математики в ігровій формі для всіх.","content5":"Отримай миттєву допомогу з математики! Замов 15-хвилинне відеозаняття з репетитором.","header1":"Стань майстром математичного вишколу","header2":"Зруш з місця","header3":"Виграй iPad та інші чудові призи ","header4":"Спробуй Bootcamps","header5":"Відеочат ","subject1":"Дроби","title":"Новини"},"newsInHomebase":"Новини на домашній базі","metaDescription":"Уся необхідна тобі допомога з математики безплатно! Отримай повний навчальний план із математики разом із навчальними відео, інтерактивним розв’язуванням завдань, іграми та цілодобовим чатом.","takeQuiz":"Пройти тест","tipOftheDay":"ПОРАДА ДНЯ","doMath":"Зайнятися математикою","doMathLT":"Займайся математикою!","learningPath":"Шлях навчання","learningPathTutorial":{"step1":{"title":"Ми створили твій \u003cspan\u003eшлях навчання\u003c/span\u003e на підставі твоїх відповідей.","action":"Далі"},"step2":{"title":"Щоб пройти рівень, переглянь \u003cspan\u003eвідео\u003c/span\u003e та виконай \u003cspan\u003eзавдання\u003c/span\u003e","action":"Дізнатися більше"},"step3":{"title":"Оціни свої знання,\u003cbr /\u003e взявши участь у \u003cspan\u003eбитві з босом\u003c/span\u003e","action":"Далі!"},"step4":{"title":"Ви завжди можете отримати нові завдання, оновивши ваш \u003cspan\u003eшлях навчання\u003c/span\u003e.","action":"Далі"},"step5":{"title":"Щоб ознайомитися з усіма нашими матеріалами, відвідай розділ \u003cspan\u003eУся математика\u003c/span\u003e","action":"Закінчити"}},"curriculumTutorial1":{"step1":{"title":"Кожна академічна шапочка \u003cCapImage /\u003e позначає математичну тему. 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Students in Texas, Virginia, Alaska, and Nebraska can also achieve strong learning outcomes with House of Math, but should use the our AI Oracle to ensure they cover all required topics. The platform supports all K–12 math globally, as well as selected university-level subjects.","cancelAnytime":"Yes. You can cancel whenever you want, and you'll keep access until the end of your current billing period. No hidden fees.","gradeLevelsCover":"We cover the full K–12 U.S. curriculum. Lessons are reviewed and verified by expert educators.","freeTrial":"Yes, we offer a 7 day free trial. Cancel anytime during the trial and you won't be charged."}}}},"getStartedV2":{"freeTakes90Seconds":"Free · Takes 90 seconds","understandChildMath":"Let's understand what's happening with your child’s math","answerQuestions":"Answer a few quick questions about what you're seeing at home. 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The real cause is usually a few missing foundational blocks that quietly make every lesson feel scary. House of Math finds exactly which blocks need rebuilding.","ES_EMO_NEW":"Seeing your child shrink away from math is heartbreaking, especially this young. When {{breakdown}} starts to feel scary, it often traces back to a few foundational blocks that never fully solidified — not to lack of effort. House of Math finds exactly which blocks are missing and rebuilds them.","ES_ACA_EXP":"You've already invested in outside support — yet progress in {{breakdown}} hasn't matched the effort. More hours rarely fixes this; finding the specific foundational blocks your child skipped does. House of Math identifies exactly those gaps and rebuilds them.","ES_ACA_NEW":"Early math foundations shape everything that comes later, so wanting your child to do well now is exactly the right instinct. When {{breakdown}} feels harder than it should, the cause is almost never effort — it's a few foundational blocks that didn't fully land. House of Math finds and rebuilds exactly those blocks.","ES_DIA_EXP":"You can tell something isn't working — even after trying outside help — but pinpointing what is harder than people admit. At this age, vague struggles almost always trace back to a few specific foundational blocks, often around {{breakdown}}. House of Math diagnoses exactly which blocks need attention and rebuilds them.","ES_DIA_NEW":"Sensing something is off — without knowing exactly what — is actually the most useful starting point. At the elementary level, vague struggles almost always come down to a few specific foundational blocks, often around {{breakdown}}. House of Math diagnoses exactly which blocks need rebuilding.","MS_EMO_EXP":"Middle school is when math anxiety hits hardest, and the extra support you've given hasn't fully reached the stress around {{breakdown}}. The pressure usually isn't about middle school content itself — it builds on earlier foundational blocks that never fully solidified. House of Math finds and rebuilds exactly those blocks, so confidence can return.","MS_EMO_NEW":"Math pressure peaks in middle school, and seeing your child shut down around {{breakdown}} is genuinely hard to watch. This anxiety almost never starts in middle school — it builds on earlier foundational blocks that quietly never solidified. House of Math finds those exact missing blocks and rebuilds them, clearing the anxiety at its source.","MS_ACA_EXP":"You've invested real time and outside support in your child's math — yet {{breakdown}} still feels harder than it should. That's not a sign of insufficient effort; it's a signal that earlier foundational blocks have gaps. House of Math finds exactly which blocks are missing and rebuilds them, so the hours invested finally translate into progress.","MS_ACA_NEW":"Middle school is a critical window — what gets solid now shapes high school and beyond. If {{breakdown}} is harder than it should be, the cause is usually a few earlier foundational blocks that didn't fully form, not effort. House of Math finds exactly those blocks and rebuilds them before middle school content can fully click.","MS_DIA_EXP":"You've tried outside help, yet something still feels off — even if you can't name exactly what. Middle school struggles almost always trace back to specific foundational blocks from earlier years, often surfacing around {{breakdown}}. House of Math diagnoses exactly which blocks are missing and rebuilds them.","MS_DIA_NEW":"Knowing something is off — without being able to pinpoint what — is actually a clear-eyed read. Middle school difficulties almost always trace back to specific foundational blocks from earlier years, often around {{breakdown}}. House of Math diagnoses exactly which blocks need rebuilding before they compound further.","HS_EMO_EXP":"High school math anxiety is uniquely heavy — grades feel permanent, and the extra support you've provided hasn't eased the stress around {{breakdown}}. High school content sits on years of earlier foundational blocks, and gaps from before are usually what's driving the anxiety now. House of Math finds those exact missing blocks and rebuilds them.","HS_EMO_NEW":"By high school, math anxiety carries real weight — grades and college pressure feed it, especially around {{breakdown}}. Pushing harder rarely helps now; the anxiety usually traces back to foundational blocks from earlier years that quietly never solidified. House of Math finds those exact blocks and rebuilds them, clearing the source of the stress.","HS_ACA_EXP":"You've invested seriously in outside support — yet {{breakdown}} still feels harder than the effort warrants. With high school stakes this high, that gap is worth taking seriously: it almost always traces to earlier foundational blocks that never fully solidified. House of Math finds those exact blocks and rebuilds them, so your child's effort finally compounds.","HS_ACA_NEW":"High school math determines real outcomes — GPA, college options. If {{breakdown}} is harder than it should be, the cause is rarely effort; high school content stacks on years of earlier foundational blocks, and gaps from before make advanced topics feel impossible. House of Math finds and rebuilds those missing blocks before the gap widens.","HS_DIA_EXP":"Knowing something is off — despite the outside help you've sought — is a more accurate read than most parents credit themselves for. 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