Se da la desigualdad:
∣ sin ( 3 x ) ∣ ≤ 2 2 \left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2} ∣ sin ( 3 x ) ∣ ≤ 2 2 Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
∣ sin ( 3 x ) ∣ = 2 2 \left|{\sin{\left(3 x \right)}}\right| = \frac{\sqrt{2}}{2} ∣ sin ( 3 x ) ∣ = 2 2 Resolvemos:
Tenemos la ecuación
∣ sin ( 3 x ) ∣ = 2 2 \left|{\sin{\left(3 x \right)}}\right| = \frac{\sqrt{2}}{2} ∣ sin ( 3 x ) ∣ = 2 2 cambiamos
∣ sin ( 3 x ) ∣ − 2 2 = 0 \left|{\sin{\left(3 x \right)}}\right| - \frac{\sqrt{2}}{2} = 0 ∣ sin ( 3 x ) ∣ − 2 2 = 0 ∣ sin ( 3 x ) ∣ − 2 2 = 0 \left|{\sin{\left(3 x \right)}}\right| - \frac{\sqrt{2}}{2} = 0 ∣ sin ( 3 x ) ∣ − 2 2 = 0 Sustituimos
w = ∣ sin ( 3 x ) ∣ w = \left|{\sin{\left(3 x \right)}}\right| w = ∣ sin ( 3 x ) ∣ Abrimos los paréntesis en el miembro izquierdo de la ecuación
w - sqrt2/2 = 0 Dividamos ambos miembros de la ecuación en (w - sqrt(2)/2)/w
w = 0 / ((w - sqrt(2)/2)/w) Obtenemos la respuesta: w = sqrt(2)/2
hacemos cambio inverso
∣ sin ( 3 x ) ∣ = w \left|{\sin{\left(3 x \right)}}\right| = w ∣ sin ( 3 x ) ∣ = w sustituimos w:
x 1 = 56.2868683768171 x_{1} = 56.2868683768171 x 1 = 56.2868683768171 x 2 = − 87.7027949127151 x_{2} = -87.7027949127151 x 2 = − 87.7027949127151 x 3 = − 71.9948316447661 x_{3} = -71.9948316447661 x 3 = − 71.9948316447661 x 4 = 18.0641577581413 x_{4} = 18.0641577581413 x 4 = 18.0641577581413 x 5 = − 6.02138591938044 x_{5} = -6.02138591938044 x 5 = − 6.02138591938044 x 6 = 25.9181393921158 x_{6} = 25.9181393921158 x 6 = 25.9181393921158 x 7 = − 65.7116463375865 x_{7} = -65.7116463375865 x 7 = − 65.7116463375865 x 8 = 40.0553063332699 x_{8} = 40.0553063332699 x 8 = 40.0553063332699 x 9 = − 47.9092879672443 x_{9} = -47.9092879672443 x 9 = − 47.9092879672443 x 10 = − 19.1113553093379 x_{10} = -19.1113553093379 x 10 = − 19.1113553093379 x 11 = 44.2440965380563 x_{11} = 44.2440965380563 x 11 = 44.2440965380563 x 12 = − 15.9697626557481 x_{12} = -15.9697626557481 x 12 = − 15.9697626557481 x 13 = − 43.720497762458 x_{13} = -43.720497762458 x 13 = − 43.720497762458 x 14 = − 62.0464549083984 x_{14} = -62.0464549083984 x 14 = − 62.0464549083984 x 15 = − 53.6688744988256 x_{15} = -53.6688744988256 x 15 = − 53.6688744988256 x 16 = − 3.92699081698724 x_{16} = -3.92699081698724 x 16 = − 3.92699081698724 x 17 = 88.2263936883134 x_{17} = 88.2263936883134 x 17 = 88.2263936883134 x 18 = − 84.037603483527 x_{18} = -84.037603483527 x 18 = − 84.037603483527 x 19 = − 31.6777259236971 x_{19} = -31.6777259236971 x 19 = − 31.6777259236971 x 20 = − 81.9432083811338 x_{20} = -81.9432083811338 x 20 = − 81.9432083811338 x 21 = 74.0892267471593 x_{21} = 74.0892267471593 x 21 = 74.0892267471593 x 22 = − 28.012534494509 x_{22} = -28.012534494509 x 22 = − 28.012534494509 x 23 = 91.8915851175014 x_{23} = 91.8915851175014 x 23 = 91.8915851175014 x 24 = − 13.8753675533549 x_{24} = -13.8753675533549 x 24 = − 13.8753675533549 x 25 = − 172.525796559639 x_{25} = -172.525796559639 x 25 = − 172.525796559639 x 26 = 0.261799387799149 x_{26} = 0.261799387799149 x 26 = 0.261799387799149 x 27 = 32.2013246992954 x_{27} = 32.2013246992954 x 27 = 32.2013246992954 x 28 = 100.269165527074 x_{28} = 100.269165527074 x 28 = 100.269165527074 x 29 = − 93.9859802198946 x_{29} = -93.9859802198946 x 29 = − 93.9859802198946 x 30 = 3.92699081698724 x_{30} = 3.92699081698724 x 30 = 3.92699081698724 x 31 = 2.35619449019234 x_{31} = 2.35619449019234 x 31 = 2.35619449019234 x 32 = − 75.6600230739542 x_{32} = -75.6600230739542 x 32 = − 75.6600230739542 x 33 = 76.1836218495525 x_{33} = 76.1836218495525 x 33 = 76.1836218495525 x 34 = − 59.9520598060052 x_{34} = -59.9520598060052 x 34 = − 59.9520598060052 x 35 = − 37.9609112308767 x_{35} = -37.9609112308767 x 35 = − 37.9609112308767 x 36 = − 9.68657734856853 x_{36} = -9.68657734856853 x 36 = − 9.68657734856853 x 37 = 46.3384916404494 x_{37} = 46.3384916404494 x 37 = 46.3384916404494 x 38 = 64.1408500107916 x_{38} = 64.1408500107916 x 38 = 64.1408500107916 x 39 = 78.2780169519457 x_{39} = 78.2780169519457 x 39 = 78.2780169519457 x 40 = 47.9092879672443 x_{40} = 47.9092879672443 x 40 = 47.9092879672443 x 41 = 1738.08613559855 x_{41} = 1738.08613559855 x 41 = 1738.08613559855 x 42 = − 91.8915851175014 x_{42} = -91.8915851175014 x 42 = − 91.8915851175014 x 43 = 42.1497014356631 x_{43} = 42.1497014356631 x 43 = 42.1497014356631 x 44 = − 50.0036830696375 x_{44} = -50.0036830696375 x 44 = − 50.0036830696375 x 45 = 96.0803753222878 x_{45} = 96.0803753222878 x 45 = 96.0803753222878 x 46 = − 33.7721210260903 x_{46} = -33.7721210260903 x 46 = − 33.7721210260903 x 47 = 71.9948316447661 x_{47} = 71.9948316447661 x 47 = 71.9948316447661 x 48 = 86.1319985859202 x_{48} = 86.1319985859202 x 48 = 86.1319985859202 x 49 = − 74.0892267471593 x_{49} = -74.0892267471593 x 49 = − 74.0892267471593 x 50 = 66.2352451131848 x_{50} = 66.2352451131848 x 50 = 66.2352451131848 x 51 = 28.012534494509 x_{51} = 28.012534494509 x 51 = 28.012534494509 x 52 = 12.30457122656 x_{52} = 12.30457122656 x 52 = 12.30457122656 x 53 = 50.0036830696375 x_{53} = 50.0036830696375 x 53 = 50.0036830696375 x 54 = 84.037603483527 x_{54} = 84.037603483527 x 54 = 84.037603483527 x 55 = 136.397481043357 x_{55} = 136.397481043357 x 55 = 136.397481043357 x 56 = 22.2529479629277 x_{56} = 22.2529479629277 x 56 = 22.2529479629277 x 57 = − 21.7293491873294 x_{57} = -21.7293491873294 x 57 = − 21.7293491873294 x 58 = − 52.0980781720307 x_{58} = -52.0980781720307 x 58 = − 52.0980781720307 x 59 = 37.9609112308767 x_{59} = 37.9609112308767 x 59 = 37.9609112308767 x 60 = 68.329640215578 x_{60} = 68.329640215578 x 60 = 68.329640215578 x 61 = 6.02138591938044 x_{61} = 6.02138591938044 x 61 = 6.02138591938044 x 62 = 59.9520598060052 x_{62} = 59.9520598060052 x 62 = 59.9520598060052 x 63 = − 23.8237442897226 x_{63} = -23.8237442897226 x 63 = − 23.8237442897226 x 64 = − 11.7809724509617 x_{64} = -11.7809724509617 x 64 = − 11.7809724509617 x 65 = − 79.8488132787406 x_{65} = -79.8488132787406 x 65 = − 79.8488132787406 x 66 = − 67.8060414399797 x_{66} = -67.8060414399797 x 66 = − 67.8060414399797 x 67 = 8.11578102177363 x_{67} = 8.11578102177363 x 67 = 8.11578102177363 x 68 = 62.0464549083984 x_{68} = 62.0464549083984 x 68 = 62.0464549083984 x 69 = − 25.9181393921158 x_{69} = -25.9181393921158 x 69 = − 25.9181393921158 x 70 = 20.1585528605345 x_{70} = 20.1585528605345 x 70 = 20.1585528605345 x 71 = − 96.0803753222878 x_{71} = -96.0803753222878 x 71 = − 96.0803753222878 x 72 = − 40.0553063332699 x_{72} = -40.0553063332699 x 72 = − 40.0553063332699 x 73 = 54.1924732744239 x_{73} = 54.1924732744239 x 73 = 54.1924732744239 x 74 = 81.9432083811338 x_{74} = 81.9432083811338 x 74 = 81.9432083811338 x 75 = 93.9859802198946 x_{75} = 93.9859802198946 x 75 = 93.9859802198946 x 76 = 34.2957198016886 x_{76} = 34.2957198016886 x 76 = 34.2957198016886 x 77 = 90.3207887907066 x_{77} = 90.3207887907066 x 77 = 90.3207887907066 x 78 = − 57.857664703612 x_{78} = -57.857664703612 x 78 = − 57.857664703612 x 79 = − 55.7632696012188 x_{79} = -55.7632696012188 x 79 = − 55.7632696012188 x 80 = 52.0980781720307 x_{80} = 52.0980781720307 x 80 = 52.0980781720307 x 81 = 81.4196096055355 x_{81} = 81.4196096055355 x 81 = 81.4196096055355 x 82 = 30.1069295969022 x_{82} = 30.1069295969022 x 82 = 30.1069295969022 x 83 = − 69.3768377667746 x_{83} = -69.3768377667746 x 83 = − 69.3768377667746 x 84 = − 89.7971900151083 x_{84} = -89.7971900151083 x 84 = − 89.7971900151083 x 85 = − 45.8148928648512 x_{85} = -45.8148928648512 x 85 = − 45.8148928648512 x 86 = − 99.7455667514759 x_{86} = -99.7455667514759 x 86 = − 99.7455667514759 x 87 = − 18.0641577581413 x_{87} = -18.0641577581413 x 87 = − 18.0641577581413 x 88 = − 1.83259571459405 x_{88} = -1.83259571459405 x 88 = − 1.83259571459405 x 89 = 15.9697626557481 x_{89} = 15.9697626557481 x 89 = 15.9697626557481 x 90 = − 77.7544181763474 x_{90} = -77.7544181763474 x 90 = − 77.7544181763474 x 91 = 10.2101761241668 x_{91} = 10.2101761241668 x 91 = 10.2101761241668 x 92 = 98.174770424681 x_{92} = 98.174770424681 x 92 = 98.174770424681 x 93 = − 69.9004365423729 x_{93} = -69.9004365423729 x 93 = − 69.9004365423729 x 94 = 69.9004365423729 x_{94} = 69.9004365423729 x 94 = 69.9004365423729 x 95 = − 35.8665161284835 x_{95} = -35.8665161284835 x 95 = − 35.8665161284835 x 96 = 24.3473430653209 x_{96} = 24.3473430653209 x 96 = 24.3473430653209 x 1 = 56.2868683768171 x_{1} = 56.2868683768171 x 1 = 56.2868683768171 x 2 = − 87.7027949127151 x_{2} = -87.7027949127151 x 2 = − 87.7027949127151 x 3 = − 71.9948316447661 x_{3} = -71.9948316447661 x 3 = − 71.9948316447661 x 4 = 18.0641577581413 x_{4} = 18.0641577581413 x 4 = 18.0641577581413 x 5 = − 6.02138591938044 x_{5} = -6.02138591938044 x 5 = − 6.02138591938044 x 6 = 25.9181393921158 x_{6} = 25.9181393921158 x 6 = 25.9181393921158 x 7 = − 65.7116463375865 x_{7} = -65.7116463375865 x 7 = − 65.7116463375865 x 8 = 40.0553063332699 x_{8} = 40.0553063332699 x 8 = 40.0553063332699 x 9 = − 47.9092879672443 x_{9} = -47.9092879672443 x 9 = − 47.9092879672443 x 10 = − 19.1113553093379 x_{10} = -19.1113553093379 x 10 = − 19.1113553093379 x 11 = 44.2440965380563 x_{11} = 44.2440965380563 x 11 = 44.2440965380563 x 12 = − 15.9697626557481 x_{12} = -15.9697626557481 x 12 = − 15.9697626557481 x 13 = − 43.720497762458 x_{13} = -43.720497762458 x 13 = − 43.720497762458 x 14 = − 62.0464549083984 x_{14} = -62.0464549083984 x 14 = − 62.0464549083984 x 15 = − 53.6688744988256 x_{15} = -53.6688744988256 x 15 = − 53.6688744988256 x 16 = − 3.92699081698724 x_{16} = -3.92699081698724 x 16 = − 3.92699081698724 x 17 = 88.2263936883134 x_{17} = 88.2263936883134 x 17 = 88.2263936883134 x 18 = − 84.037603483527 x_{18} = -84.037603483527 x 18 = − 84.037603483527 x 19 = − 31.6777259236971 x_{19} = -31.6777259236971 x 19 = − 31.6777259236971 x 20 = − 81.9432083811338 x_{20} = -81.9432083811338 x 20 = − 81.9432083811338 x 21 = 74.0892267471593 x_{21} = 74.0892267471593 x 21 = 74.0892267471593 x 22 = − 28.012534494509 x_{22} = -28.012534494509 x 22 = − 28.012534494509 x 23 = 91.8915851175014 x_{23} = 91.8915851175014 x 23 = 91.8915851175014 x 24 = − 13.8753675533549 x_{24} = -13.8753675533549 x 24 = − 13.8753675533549 x 25 = − 172.525796559639 x_{25} = -172.525796559639 x 25 = − 172.525796559639 x 26 = 0.261799387799149 x_{26} = 0.261799387799149 x 26 = 0.261799387799149 x 27 = 32.2013246992954 x_{27} = 32.2013246992954 x 27 = 32.2013246992954 x 28 = 100.269165527074 x_{28} = 100.269165527074 x 28 = 100.269165527074 x 29 = − 93.9859802198946 x_{29} = -93.9859802198946 x 29 = − 93.9859802198946 x 30 = 3.92699081698724 x_{30} = 3.92699081698724 x 30 = 3.92699081698724 x 31 = 2.35619449019234 x_{31} = 2.35619449019234 x 31 = 2.35619449019234 x 32 = − 75.6600230739542 x_{32} = -75.6600230739542 x 32 = − 75.6600230739542 x 33 = 76.1836218495525 x_{33} = 76.1836218495525 x 33 = 76.1836218495525 x 34 = − 59.9520598060052 x_{34} = -59.9520598060052 x 34 = − 59.9520598060052 x 35 = − 37.9609112308767 x_{35} = -37.9609112308767 x 35 = − 37.9609112308767 x 36 = − 9.68657734856853 x_{36} = -9.68657734856853 x 36 = − 9.68657734856853 x 37 = 46.3384916404494 x_{37} = 46.3384916404494 x 37 = 46.3384916404494 x 38 = 64.1408500107916 x_{38} = 64.1408500107916 x 38 = 64.1408500107916 x 39 = 78.2780169519457 x_{39} = 78.2780169519457 x 39 = 78.2780169519457 x 40 = 47.9092879672443 x_{40} = 47.9092879672443 x 40 = 47.9092879672443 x 41 = 1738.08613559855 x_{41} = 1738.08613559855 x 41 = 1738.08613559855 x 42 = − 91.8915851175014 x_{42} = -91.8915851175014 x 42 = − 91.8915851175014 x 43 = 42.1497014356631 x_{43} = 42.1497014356631 x 43 = 42.1497014356631 x 44 = − 50.0036830696375 x_{44} = -50.0036830696375 x 44 = − 50.0036830696375 x 45 = 96.0803753222878 x_{45} = 96.0803753222878 x 45 = 96.0803753222878 x 46 = − 33.7721210260903 x_{46} = -33.7721210260903 x 46 = − 33.7721210260903 x 47 = 71.9948316447661 x_{47} = 71.9948316447661 x 47 = 71.9948316447661 x 48 = 86.1319985859202 x_{48} = 86.1319985859202 x 48 = 86.1319985859202 x 49 = − 74.0892267471593 x_{49} = -74.0892267471593 x 49 = − 74.0892267471593 x 50 = 66.2352451131848 x_{50} = 66.2352451131848 x 50 = 66.2352451131848 x 51 = 28.012534494509 x_{51} = 28.012534494509 x 51 = 28.012534494509 x 52 = 12.30457122656 x_{52} = 12.30457122656 x 52 = 12.30457122656 x 53 = 50.0036830696375 x_{53} = 50.0036830696375 x 53 = 50.0036830696375 x 54 = 84.037603483527 x_{54} = 84.037603483527 x 54 = 84.037603483527 x 55 = 136.397481043357 x_{55} = 136.397481043357 x 55 = 136.397481043357 x 56 = 22.2529479629277 x_{56} = 22.2529479629277 x 56 = 22.2529479629277 x 57 = − 21.7293491873294 x_{57} = -21.7293491873294 x 57 = − 21.7293491873294 x 58 = − 52.0980781720307 x_{58} = -52.0980781720307 x 58 = − 52.0980781720307 x 59 = 37.9609112308767 x_{59} = 37.9609112308767 x 59 = 37.9609112308767 x 60 = 68.329640215578 x_{60} = 68.329640215578 x 60 = 68.329640215578 x 61 = 6.02138591938044 x_{61} = 6.02138591938044 x 61 = 6.02138591938044 x 62 = 59.9520598060052 x_{62} = 59.9520598060052 x 62 = 59.9520598060052 x 63 = − 23.8237442897226 x_{63} = -23.8237442897226 x 63 = − 23.8237442897226 x 64 = − 11.7809724509617 x_{64} = -11.7809724509617 x 64 = − 11.7809724509617 x 65 = − 79.8488132787406 x_{65} = -79.8488132787406 x 65 = − 79.8488132787406 x 66 = − 67.8060414399797 x_{66} = -67.8060414399797 x 66 = − 67.8060414399797 x 67 = 8.11578102177363 x_{67} = 8.11578102177363 x 67 = 8.11578102177363 x 68 = 62.0464549083984 x_{68} = 62.0464549083984 x 68 = 62.0464549083984 x 69 = − 25.9181393921158 x_{69} = -25.9181393921158 x 69 = − 25.9181393921158 x 70 = 20.1585528605345 x_{70} = 20.1585528605345 x 70 = 20.1585528605345 x 71 = − 96.0803753222878 x_{71} = -96.0803753222878 x 71 = − 96.0803753222878 x 72 = − 40.0553063332699 x_{72} = -40.0553063332699 x 72 = − 40.0553063332699 x 73 = 54.1924732744239 x_{73} = 54.1924732744239 x 73 = 54.1924732744239 x 74 = 81.9432083811338 x_{74} = 81.9432083811338 x 74 = 81.9432083811338 x 75 = 93.9859802198946 x_{75} = 93.9859802198946 x 75 = 93.9859802198946 x 76 = 34.2957198016886 x_{76} = 34.2957198016886 x 76 = 34.2957198016886 x 77 = 90.3207887907066 x_{77} = 90.3207887907066 x 77 = 90.3207887907066 x 78 = − 57.857664703612 x_{78} = -57.857664703612 x 78 = − 57.857664703612 x 79 = − 55.7632696012188 x_{79} = -55.7632696012188 x 79 = − 55.7632696012188 x 80 = 52.0980781720307 x_{80} = 52.0980781720307 x 80 = 52.0980781720307 x 81 = 81.4196096055355 x_{81} = 81.4196096055355 x 81 = 81.4196096055355 x 82 = 30.1069295969022 x_{82} = 30.1069295969022 x 82 = 30.1069295969022 x 83 = − 69.3768377667746 x_{83} = -69.3768377667746 x 83 = − 69.3768377667746 x 84 = − 89.7971900151083 x_{84} = -89.7971900151083 x 84 = − 89.7971900151083 x 85 = − 45.8148928648512 x_{85} = -45.8148928648512 x 85 = − 45.8148928648512 x 86 = − 99.7455667514759 x_{86} = -99.7455667514759 x 86 = − 99.7455667514759 x 87 = − 18.0641577581413 x_{87} = -18.0641577581413 x 87 = − 18.0641577581413 x 88 = − 1.83259571459405 x_{88} = -1.83259571459405 x 88 = − 1.83259571459405 x 89 = 15.9697626557481 x_{89} = 15.9697626557481 x 89 = 15.9697626557481 x 90 = − 77.7544181763474 x_{90} = -77.7544181763474 x 90 = − 77.7544181763474 x 91 = 10.2101761241668 x_{91} = 10.2101761241668 x 91 = 10.2101761241668 x 92 = 98.174770424681 x_{92} = 98.174770424681 x 92 = 98.174770424681 x 93 = − 69.9004365423729 x_{93} = -69.9004365423729 x 93 = − 69.9004365423729 x 94 = 69.9004365423729 x_{94} = 69.9004365423729 x 94 = 69.9004365423729 x 95 = − 35.8665161284835 x_{95} = -35.8665161284835 x 95 = − 35.8665161284835 x 96 = 24.3473430653209 x_{96} = 24.3473430653209 x 96 = 24.3473430653209 Las raíces dadas
x 25 = − 172.525796559639 x_{25} = -172.525796559639 x 25 = − 172.525796559639 x 86 = − 99.7455667514759 x_{86} = -99.7455667514759 x 86 = − 99.7455667514759 x 71 = − 96.0803753222878 x_{71} = -96.0803753222878 x 71 = − 96.0803753222878 x 29 = − 93.9859802198946 x_{29} = -93.9859802198946 x 29 = − 93.9859802198946 x 42 = − 91.8915851175014 x_{42} = -91.8915851175014 x 42 = − 91.8915851175014 x 84 = − 89.7971900151083 x_{84} = -89.7971900151083 x 84 = − 89.7971900151083 x 2 = − 87.7027949127151 x_{2} = -87.7027949127151 x 2 = − 87.7027949127151 x 18 = − 84.037603483527 x_{18} = -84.037603483527 x 18 = − 84.037603483527 x 20 = − 81.9432083811338 x_{20} = -81.9432083811338 x 20 = − 81.9432083811338 x 65 = − 79.8488132787406 x_{65} = -79.8488132787406 x 65 = − 79.8488132787406 x 90 = − 77.7544181763474 x_{90} = -77.7544181763474 x 90 = − 77.7544181763474 x 32 = − 75.6600230739542 x_{32} = -75.6600230739542 x 32 = − 75.6600230739542 x 49 = − 74.0892267471593 x_{49} = -74.0892267471593 x 49 = − 74.0892267471593 x 3 = − 71.9948316447661 x_{3} = -71.9948316447661 x 3 = − 71.9948316447661 x 93 = − 69.9004365423729 x_{93} = -69.9004365423729 x 93 = − 69.9004365423729 x 83 = − 69.3768377667746 x_{83} = -69.3768377667746 x 83 = − 69.3768377667746 x 66 = − 67.8060414399797 x_{66} = -67.8060414399797 x 66 = − 67.8060414399797 x 7 = − 65.7116463375865 x_{7} = -65.7116463375865 x 7 = − 65.7116463375865 x 14 = − 62.0464549083984 x_{14} = -62.0464549083984 x 14 = − 62.0464549083984 x 34 = − 59.9520598060052 x_{34} = -59.9520598060052 x 34 = − 59.9520598060052 x 78 = − 57.857664703612 x_{78} = -57.857664703612 x 78 = − 57.857664703612 x 79 = − 55.7632696012188 x_{79} = -55.7632696012188 x 79 = − 55.7632696012188 x 15 = − 53.6688744988256 x_{15} = -53.6688744988256 x 15 = − 53.6688744988256 x 58 = − 52.0980781720307 x_{58} = -52.0980781720307 x 58 = − 52.0980781720307 x 44 = − 50.0036830696375 x_{44} = -50.0036830696375 x 44 = − 50.0036830696375 x 9 = − 47.9092879672443 x_{9} = -47.9092879672443 x 9 = − 47.9092879672443 x 85 = − 45.8148928648512 x_{85} = -45.8148928648512 x 85 = − 45.8148928648512 x 13 = − 43.720497762458 x_{13} = -43.720497762458 x 13 = − 43.720497762458 x 72 = − 40.0553063332699 x_{72} = -40.0553063332699 x 72 = − 40.0553063332699 x 35 = − 37.9609112308767 x_{35} = -37.9609112308767 x 35 = − 37.9609112308767 x 95 = − 35.8665161284835 x_{95} = -35.8665161284835 x 95 = − 35.8665161284835 x 46 = − 33.7721210260903 x_{46} = -33.7721210260903 x 46 = − 33.7721210260903 x 19 = − 31.6777259236971 x_{19} = -31.6777259236971 x 19 = − 31.6777259236971 x 22 = − 28.012534494509 x_{22} = -28.012534494509 x 22 = − 28.012534494509 x 69 = − 25.9181393921158 x_{69} = -25.9181393921158 x 69 = − 25.9181393921158 x 63 = − 23.8237442897226 x_{63} = -23.8237442897226 x 63 = − 23.8237442897226 x 57 = − 21.7293491873294 x_{57} = -21.7293491873294 x 57 = − 21.7293491873294 x 10 = − 19.1113553093379 x_{10} = -19.1113553093379 x 10 = − 19.1113553093379 x 87 = − 18.0641577581413 x_{87} = -18.0641577581413 x 87 = − 18.0641577581413 x 12 = − 15.9697626557481 x_{12} = -15.9697626557481 x 12 = − 15.9697626557481 x 24 = − 13.8753675533549 x_{24} = -13.8753675533549 x 24 = − 13.8753675533549 x 64 = − 11.7809724509617 x_{64} = -11.7809724509617 x 64 = − 11.7809724509617 x 36 = − 9.68657734856853 x_{36} = -9.68657734856853 x 36 = − 9.68657734856853 x 5 = − 6.02138591938044 x_{5} = -6.02138591938044 x 5 = − 6.02138591938044 x 16 = − 3.92699081698724 x_{16} = -3.92699081698724 x 16 = − 3.92699081698724 x 88 = − 1.83259571459405 x_{88} = -1.83259571459405 x 88 = − 1.83259571459405 x 26 = 0.261799387799149 x_{26} = 0.261799387799149 x 26 = 0.261799387799149 x 31 = 2.35619449019234 x_{31} = 2.35619449019234 x 31 = 2.35619449019234 x 30 = 3.92699081698724 x_{30} = 3.92699081698724 x 30 = 3.92699081698724 x 61 = 6.02138591938044 x_{61} = 6.02138591938044 x 61 = 6.02138591938044 x 67 = 8.11578102177363 x_{67} = 8.11578102177363 x 67 = 8.11578102177363 x 91 = 10.2101761241668 x_{91} = 10.2101761241668 x 91 = 10.2101761241668 x 52 = 12.30457122656 x_{52} = 12.30457122656 x 52 = 12.30457122656 x 89 = 15.9697626557481 x_{89} = 15.9697626557481 x 89 = 15.9697626557481 x 4 = 18.0641577581413 x_{4} = 18.0641577581413 x 4 = 18.0641577581413 x 70 = 20.1585528605345 x_{70} = 20.1585528605345 x 70 = 20.1585528605345 x 56 = 22.2529479629277 x_{56} = 22.2529479629277 x 56 = 22.2529479629277 x 96 = 24.3473430653209 x_{96} = 24.3473430653209 x 96 = 24.3473430653209 x 6 = 25.9181393921158 x_{6} = 25.9181393921158 x 6 = 25.9181393921158 x 51 = 28.012534494509 x_{51} = 28.012534494509 x 51 = 28.012534494509 x 82 = 30.1069295969022 x_{82} = 30.1069295969022 x 82 = 30.1069295969022 x 27 = 32.2013246992954 x_{27} = 32.2013246992954 x 27 = 32.2013246992954 x 76 = 34.2957198016886 x_{76} = 34.2957198016886 x 76 = 34.2957198016886 x 59 = 37.9609112308767 x_{59} = 37.9609112308767 x 59 = 37.9609112308767 x 8 = 40.0553063332699 x_{8} = 40.0553063332699 x 8 = 40.0553063332699 x 43 = 42.1497014356631 x_{43} = 42.1497014356631 x 43 = 42.1497014356631 x 11 = 44.2440965380563 x_{11} = 44.2440965380563 x 11 = 44.2440965380563 x 37 = 46.3384916404494 x_{37} = 46.3384916404494 x 37 = 46.3384916404494 x 40 = 47.9092879672443 x_{40} = 47.9092879672443 x 40 = 47.9092879672443 x 53 = 50.0036830696375 x_{53} = 50.0036830696375 x 53 = 50.0036830696375 x 80 = 52.0980781720307 x_{80} = 52.0980781720307 x 80 = 52.0980781720307 x 73 = 54.1924732744239 x_{73} = 54.1924732744239 x 73 = 54.1924732744239 x 1 = 56.2868683768171 x_{1} = 56.2868683768171 x 1 = 56.2868683768171 x 62 = 59.9520598060052 x_{62} = 59.9520598060052 x 62 = 59.9520598060052 x 68 = 62.0464549083984 x_{68} = 62.0464549083984 x 68 = 62.0464549083984 x 38 = 64.1408500107916 x_{38} = 64.1408500107916 x 38 = 64.1408500107916 x 50 = 66.2352451131848 x_{50} = 66.2352451131848 x 50 = 66.2352451131848 x 60 = 68.329640215578 x_{60} = 68.329640215578 x 60 = 68.329640215578 x 94 = 69.9004365423729 x_{94} = 69.9004365423729 x 94 = 69.9004365423729 x 47 = 71.9948316447661 x_{47} = 71.9948316447661 x 47 = 71.9948316447661 x 21 = 74.0892267471593 x_{21} = 74.0892267471593 x 21 = 74.0892267471593 x 33 = 76.1836218495525 x_{33} = 76.1836218495525 x 33 = 76.1836218495525 x 39 = 78.2780169519457 x_{39} = 78.2780169519457 x 39 = 78.2780169519457 x 81 = 81.4196096055355 x_{81} = 81.4196096055355 x 81 = 81.4196096055355 x 74 = 81.9432083811338 x_{74} = 81.9432083811338 x 74 = 81.9432083811338 x 54 = 84.037603483527 x_{54} = 84.037603483527 x 54 = 84.037603483527 x 48 = 86.1319985859202 x_{48} = 86.1319985859202 x 48 = 86.1319985859202 x 17 = 88.2263936883134 x_{17} = 88.2263936883134 x 17 = 88.2263936883134 x 77 = 90.3207887907066 x_{77} = 90.3207887907066 x 77 = 90.3207887907066 x 23 = 91.8915851175014 x_{23} = 91.8915851175014 x 23 = 91.8915851175014 x 75 = 93.9859802198946 x_{75} = 93.9859802198946 x 75 = 93.9859802198946 x 45 = 96.0803753222878 x_{45} = 96.0803753222878 x 45 = 96.0803753222878 x 92 = 98.174770424681 x_{92} = 98.174770424681 x 92 = 98.174770424681 x 28 = 100.269165527074 x_{28} = 100.269165527074 x 28 = 100.269165527074 x 55 = 136.397481043357 x_{55} = 136.397481043357 x 55 = 136.397481043357 x 41 = 1738.08613559855 x_{41} = 1738.08613559855 x 41 = 1738.08613559855 son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x 0 ≤ x 25 x_{0} \leq x_{25} x 0 ≤ x 25 Consideremos, por ejemplo, el punto
x 0 = x 25 − 1 10 x_{0} = x_{25} - \frac{1}{10} x 0 = x 25 − 10 1 =
− 172.525796559639 + − 1 10 -172.525796559639 + - \frac{1}{10} − 172.525796559639 + − 10 1 =
− 172.625796559639 -172.625796559639 − 172.625796559639 lo sustituimos en la expresión
∣ sin ( 3 x ) ∣ ≤ 2 2 \left|{\sin{\left(3 x \right)}}\right| \leq \frac{\sqrt{2}}{2} ∣ sin ( 3 x ) ∣ ≤ 2 2 ∣ sin ( ( − 172.625796559639 ) 3 ) ∣ ≤ 2 2 \left|{\sin{\left(\left(-172.625796559639\right) 3 \right)}}\right| \leq \frac{\sqrt{2}}{2} ∣ sin ( ( − 172.625796559639 ) 3 ) ∣ ≤ 2 2 ___
\/ 2
0.466560567667758 <= -----
2
significa que una de las soluciones de nuestra ecuación será con:
x ≤ − 172.525796559639 x \leq -172.525796559639 x ≤ − 172.525796559639 _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____ _____
\ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ / \ /
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x25 x86 x71 x29 x42 x84 x2 x18 x20 x65 x90 x32 x49 x3 x93 x83 x66 x7 x14 x34 x78 x79 x15 x58 x44 x9 x85 x13 x72 x35 x95 x46 x19 x22 x69 x63 x57 x10 x87 x12 x24 x64 x36 x5 x16 x88 x26 x31 x30 x61 x67 x91 x52 x89 x4 x70 x56 x96 x6 x51 x82 x27 x76 x59 x8 x43 x11 x37 x40 x53 x80 x73 x1 x62 x68 x38 x50 x60 x94 x47 x21 x33 x39 x81 x74 x54 x48 x17 x77 x23 x75 x45 x92 x28 x55 x41 Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x ≤ − 172.525796559639 x \leq -172.525796559639 x ≤ − 172.525796559639 x ≥ − 99.7455667514759 ∧ x ≤ − 96.0803753222878 x \geq -99.7455667514759 \wedge x \leq -96.0803753222878 x ≥ − 99.7455667514759 ∧ x ≤ − 96.0803753222878 x ≥ − 93.9859802198946 ∧ x ≤ − 91.8915851175014 x \geq -93.9859802198946 \wedge x \leq -91.8915851175014 x ≥ − 93.9859802198946 ∧ x ≤ − 91.8915851175014 x ≥ − 89.7971900151083 ∧ x ≤ − 87.7027949127151 x \geq -89.7971900151083 \wedge x \leq -87.7027949127151 x ≥ − 89.7971900151083 ∧ x ≤ − 87.7027949127151 x ≥ − 84.037603483527 ∧ x ≤ − 81.9432083811338 x \geq -84.037603483527 \wedge x \leq -81.9432083811338 x ≥ − 84.037603483527 ∧ x ≤ − 81.9432083811338 x ≥ − 79.8488132787406 ∧ x ≤ − 77.7544181763474 x \geq -79.8488132787406 \wedge x \leq -77.7544181763474 x ≥ − 79.8488132787406 ∧ x ≤ − 77.7544181763474 x ≥ − 75.6600230739542 ∧ x ≤ − 74.0892267471593 x \geq -75.6600230739542 \wedge x \leq -74.0892267471593 x ≥ − 75.6600230739542 ∧ x ≤ − 74.0892267471593 x ≥ − 71.9948316447661 ∧ x ≤ − 69.9004365423729 x \geq -71.9948316447661 \wedge x \leq -69.9004365423729 x ≥ − 71.9948316447661 ∧ x ≤ − 69.9004365423729 x ≥ − 69.3768377667746 ∧ x ≤ − 67.8060414399797 x \geq -69.3768377667746 \wedge x \leq -67.8060414399797 x ≥ − 69.3768377667746 ∧ x ≤ − 67.8060414399797 x ≥ − 65.7116463375865 ∧ x ≤ − 62.0464549083984 x \geq -65.7116463375865 \wedge x \leq -62.0464549083984 x ≥ − 65.7116463375865 ∧ x ≤ − 62.0464549083984 x ≥ − 59.9520598060052 ∧ x ≤ − 57.857664703612 x \geq -59.9520598060052 \wedge x \leq -57.857664703612 x ≥ − 59.9520598060052 ∧ x ≤ − 57.857664703612 x ≥ − 55.7632696012188 ∧ x ≤ − 53.6688744988256 x \geq -55.7632696012188 \wedge x \leq -53.6688744988256 x ≥ − 55.7632696012188 ∧ x ≤ − 53.6688744988256 x ≥ − 52.0980781720307 ∧ x ≤ − 50.0036830696375 x \geq -52.0980781720307 \wedge x \leq -50.0036830696375 x ≥ − 52.0980781720307 ∧ x ≤ − 50.0036830696375 x ≥ − 47.9092879672443 ∧ x ≤ − 45.8148928648512 x \geq -47.9092879672443 \wedge x \leq -45.8148928648512 x ≥ − 47.9092879672443 ∧ x ≤ − 45.8148928648512 x ≥ − 43.720497762458 ∧ x ≤ − 40.0553063332699 x \geq -43.720497762458 \wedge x \leq -40.0553063332699 x ≥ − 43.720497762458 ∧ x ≤ − 40.0553063332699 x ≥ − 37.9609112308767 ∧ x ≤ − 35.8665161284835 x \geq -37.9609112308767 \wedge x \leq -35.8665161284835 x ≥ − 37.9609112308767 ∧ x ≤ − 35.8665161284835 x ≥ − 33.7721210260903 ∧ x ≤ − 31.6777259236971 x \geq -33.7721210260903 \wedge x \leq -31.6777259236971 x ≥ − 33.7721210260903 ∧ x ≤ − 31.6777259236971 x ≥ − 28.012534494509 ∧ x ≤ − 25.9181393921158 x \geq -28.012534494509 \wedge x \leq -25.9181393921158 x ≥ − 28.012534494509 ∧ x ≤ − 25.9181393921158 x ≥ − 23.8237442897226 ∧ x ≤ − 21.7293491873294 x \geq -23.8237442897226 \wedge x \leq -21.7293491873294 x ≥ − 23.8237442897226 ∧ x ≤ − 21.7293491873294 x ≥ − 19.1113553093379 ∧ x ≤ − 18.0641577581413 x \geq -19.1113553093379 \wedge x \leq -18.0641577581413 x ≥ − 19.1113553093379 ∧ x ≤ − 18.0641577581413 x ≥ − 15.9697626557481 ∧ x ≤ − 13.8753675533549 x \geq -15.9697626557481 \wedge x \leq -13.8753675533549 x ≥ − 15.9697626557481 ∧ x ≤ − 13.8753675533549 x ≥ − 11.7809724509617 ∧ x ≤ − 9.68657734856853 x \geq -11.7809724509617 \wedge x \leq -9.68657734856853 x ≥ − 11.7809724509617 ∧ x ≤ − 9.68657734856853 x ≥ − 6.02138591938044 ∧ x ≤ − 3.92699081698724 x \geq -6.02138591938044 \wedge x \leq -3.92699081698724 x ≥ − 6.02138591938044 ∧ x ≤ − 3.92699081698724 x ≥ − 1.83259571459405 ∧ x ≤ 0.261799387799149 x \geq -1.83259571459405 \wedge x \leq 0.261799387799149 x ≥ − 1.83259571459405 ∧ x ≤ 0.261799387799149 x ≥ 2.35619449019234 ∧ x ≤ 3.92699081698724 x \geq 2.35619449019234 \wedge x \leq 3.92699081698724 x ≥ 2.35619449019234 ∧ x ≤ 3.92699081698724 x ≥ 6.02138591938044 ∧ x ≤ 8.11578102177363 x \geq 6.02138591938044 \wedge x \leq 8.11578102177363 x ≥ 6.02138591938044 ∧ x ≤ 8.11578102177363 x ≥ 10.2101761241668 ∧ x ≤ 12.30457122656 x \geq 10.2101761241668 \wedge x \leq 12.30457122656 x ≥ 10.2101761241668 ∧ x ≤ 12.30457122656 x ≥ 15.9697626557481 ∧ x ≤ 18.0641577581413 x \geq 15.9697626557481 \wedge x \leq 18.0641577581413 x ≥ 15.9697626557481 ∧ x ≤ 18.0641577581413 x ≥ 20.1585528605345 ∧ x ≤ 22.2529479629277 x \geq 20.1585528605345 \wedge x \leq 22.2529479629277 x ≥ 20.1585528605345 ∧ x ≤ 22.2529479629277 x ≥ 24.3473430653209 ∧ x ≤ 25.9181393921158 x \geq 24.3473430653209 \wedge x \leq 25.9181393921158 x ≥ 24.3473430653209 ∧ x ≤ 25.9181393921158 x ≥ 28.012534494509 ∧ x ≤ 30.1069295969022 x \geq 28.012534494509 \wedge x \leq 30.1069295969022 x ≥ 28.012534494509 ∧ x ≤ 30.1069295969022 x ≥ 32.2013246992954 ∧ x ≤ 34.2957198016886 x \geq 32.2013246992954 \wedge x \leq 34.2957198016886 x ≥ 32.2013246992954 ∧ x ≤ 34.2957198016886 x ≥ 37.9609112308767 ∧ x ≤ 40.0553063332699 x \geq 37.9609112308767 \wedge x \leq 40.0553063332699 x ≥ 37.9609112308767 ∧ x ≤ 40.0553063332699 x ≥ 42.1497014356631 ∧ x ≤ 44.2440965380563 x \geq 42.1497014356631 \wedge x \leq 44.2440965380563 x ≥ 42.1497014356631 ∧ x ≤ 44.2440965380563 x ≥ 46.3384916404494 ∧ x ≤ 47.9092879672443 x \geq 46.3384916404494 \wedge x \leq 47.9092879672443 x ≥ 46.3384916404494 ∧ x ≤ 47.9092879672443 x ≥ 50.0036830696375 ∧ x ≤ 52.0980781720307 x \geq 50.0036830696375 \wedge x \leq 52.0980781720307 x ≥ 50.0036830696375 ∧ x ≤ 52.0980781720307 x ≥ 54.1924732744239 ∧ x ≤ 56.2868683768171 x \geq 54.1924732744239 \wedge x \leq 56.2868683768171 x ≥ 54.1924732744239 ∧ x ≤ 56.2868683768171 x ≥ 59.9520598060052 ∧ x ≤ 62.0464549083984 x \geq 59.9520598060052 \wedge x \leq 62.0464549083984 x ≥ 59.9520598060052 ∧ x ≤ 62.0464549083984 x ≥ 64.1408500107916 ∧ x ≤ 66.2352451131848 x \geq 64.1408500107916 \wedge x \leq 66.2352451131848 x ≥ 64.1408500107916 ∧ x ≤ 66.2352451131848 x ≥ 68.329640215578 ∧ x ≤ 69.9004365423729 x \geq 68.329640215578 \wedge x \leq 69.9004365423729 x ≥ 68.329640215578 ∧ x ≤ 69.9004365423729 x ≥ 71.9948316447661 ∧ x ≤ 74.0892267471593 x \geq 71.9948316447661 \wedge x \leq 74.0892267471593 x ≥ 71.9948316447661 ∧ x ≤ 74.0892267471593 x ≥ 76.1836218495525 ∧ x ≤ 78.2780169519457 x \geq 76.1836218495525 \wedge x \leq 78.2780169519457 x ≥ 76.1836218495525 ∧ x ≤ 78.2780169519457 x ≥ 81.4196096055355 ∧ x ≤ 81.9432083811338 x \geq 81.4196096055355 \wedge x \leq 81.9432083811338 x ≥ 81.4196096055355 ∧ x ≤ 81.9432083811338 x ≥ 84.037603483527 ∧ x ≤ 86.1319985859202 x \geq 84.037603483527 \wedge x \leq 86.1319985859202 x ≥ 84.037603483527 ∧ x ≤ 86.1319985859202 x ≥ 88.2263936883134 ∧ x ≤ 90.3207887907066 x \geq 88.2263936883134 \wedge x \leq 90.3207887907066 x ≥ 88.2263936883134 ∧ x ≤ 90.3207887907066 x ≥ 91.8915851175014 ∧ x ≤ 93.9859802198946 x \geq 91.8915851175014 \wedge x \leq 93.9859802198946 x ≥ 91.8915851175014 ∧ x ≤ 93.9859802198946 x ≥ 96.0803753222878 ∧ x ≤ 98.174770424681 x \geq 96.0803753222878 \wedge x \leq 98.174770424681 x ≥ 96.0803753222878 ∧ x ≤ 98.174770424681 x ≥ 100.269165527074 ∧ x ≤ 136.397481043357 x \geq 100.269165527074 \wedge x \leq 136.397481043357 x ≥ 100.269165527074 ∧ x ≤ 136.397481043357 x ≥ 1738.08613559855 x \geq 1738.08613559855 x ≥ 1738.08613559855 
x^2+15x>0
x^2+3x>0
x^2+15>0