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Resolver desigualdades/ sqrt((x+2)-4*sqrt(x-2))+sqrt(x+7-6*sqrt(x-2))>=1

sqrt((x+2)-4*sqrt(x-2))+sqrt(x+7-6*sqrt(x-2))>=1 desigualdades

En la desigualdad la incógnita

Solución

Ha introducido [src] 
   _____________________      _____________________     
  /             _______      /             _______      
\/  x + 2 - 4*\/ x - 2   + \/  x + 7 - 6*\/ x - 2   >= 1
−6x−2+(x+7)+−4x−2+(x+2)≥1\sqrt{- 6 \sqrt{x - 2} + \left(x + 7\right)} + \sqrt{- 4 \sqrt{x - 2} + \left(x + 2\right)} \geq 1 
sqrt(-6*sqrt(x - 2) + x + 7) + sqrt(-4*sqrt(x - 2) + x + 2) >= 1
Solución detallada
Se da la desigualdad:
−6x−2+(x+7)+−4x−2+(x+2)≥1\sqrt{- 6 \sqrt{x - 2} + \left(x + 7\right)} + \sqrt{- 4 \sqrt{x - 2} + \left(x + 2\right)} \geq 1
Para resolver esta desigualdad primero hay que resolver la ecuación correspondiente:
−6x−2+(x+7)+−4x−2+(x+2)=1\sqrt{- 6 \sqrt{x - 2} + \left(x + 7\right)} + \sqrt{- 4 \sqrt{x - 2} + \left(x + 2\right)} = 1
Resolvemos:
x1=−209.185730465882−69.3869132967511ix_{1} = -209.185730465882 - 69.3869132967511 i
x2=−16.3316209432957+25.2384811072174ix_{2} = -16.3316209432957 + 25.2384811072174 i
x3=2.97429800224351+10.5816945161317ix_{3} = 2.97429800224351 + 10.5816945161317 i
x4=9.60642720282926−2.01436290069231ix_{4} = 9.60642720282926 - 2.01436290069231 i
x5=6.80426490214766+1.72764871868375ix_{5} = 6.80426490214766 + 1.72764871868375 i
x6=3.39083443957248+12.9050220965973ix_{6} = 3.39083443957248 + 12.9050220965973 i
x7=9.41935871894229x_{7} = 9.41935871894229
x8=−157.836209650189−62.7058591221354ix_{8} = -157.836209650189 - 62.7058591221354 i
x9=8.25x_{9} = 8.25
x10=−598.199875646853+128.092404329392ix_{10} = -598.199875646853 + 128.092404329392 i
x11=−17.1821239898267+28.6263470704165ix_{11} = -17.1821239898267 + 28.6263470704165 i
x12=−602.961210517373+117.235765419086ix_{12} = -602.961210517373 + 117.235765419086 i
x13=−464.751570442102+97.1690141907255ix_{13} = -464.751570442102 + 97.1690141907255 i
x14=1.85813942443044−17.96102403561ix_{14} = 1.85813942443044 - 17.96102403561 i
x15=−11.703200456428−27.4255414085721ix_{15} = -11.703200456428 - 27.4255414085721 i
x16=7.23060661206161−1.99103861084522ix_{16} = 7.23060661206161 - 1.99103861084522 i
x17=−596.008821403641+135.432924067879ix_{17} = -596.008821403641 + 135.432924067879 i
x18=7.76367700154925+2.85546175661114ix_{18} = 7.76367700154925 + 2.85546175661114 i
x19=−9.83261157058662−26.0573597106802ix_{19} = -9.83261157058662 - 26.0573597106802 i
x20=0.0889057621612147−19.0875236891862ix_{20} = 0.0889057621612147 - 19.0875236891862 i
x21=−465.492577721658+87.3262017116185ix_{21} = -465.492577721658 + 87.3262017116185 i
x22=−604.934157360214+113.834101906546ix_{22} = -604.934157360214 + 113.834101906546 i
x23=7.50920203463435+2.38511851816894ix_{23} = 7.50920203463435 + 2.38511851816894 i
x24=−8.85289417928458+20.9387849888536ix_{24} = -8.85289417928458 + 20.9387849888536 i
x25=−15.6712269874307+19.2501813678031ix_{25} = -15.6712269874307 + 19.2501813678031 i
x26=−122.512959882748+62.1332260804273ix_{26} = -122.512959882748 + 62.1332260804273 i
x27=−15.8657958924392+28.1625848403159ix_{27} = -15.8657958924392 + 28.1625848403159 i
x28=7.33854043758934−2.81069661981164ix_{28} = 7.33854043758934 - 2.81069661981164 i
x29=−464.86074523675+93.939436524962ix_{29} = -464.86074523675 + 93.939436524962 i
x30=−0.419634792565839−12.7472942155552ix_{30} = -0.419634792565839 - 12.7472942155552 i
x31=8.09757208771761x_{31} = 8.09757208771761
x32=−3.45866753605209−21.4923203402758ix_{32} = -3.45866753605209 - 21.4923203402758 i
x33=−607.062717462228+110.68252768482ix_{33} = -607.062717462228 + 110.68252768482 i
x34=−15.3320513491424+22.7906906901023ix_{34} = -15.3320513491424 + 22.7906906901023 i
x35=−5.25396678065631−22.7562739262181ix_{35} = -5.25396678065631 - 22.7562739262181 i
x36=−18.2536905544404+31.784077340187ix_{36} = -18.2536905544404 + 31.784077340187 i
x37=−7.06908033916105−24.0543783710452ix_{37} = -7.06908033916105 - 24.0543783710452 i
x38=3.79638502210314+15.2324038414955ix_{38} = 3.79638502210314 + 15.2324038414955 i
x39=7.34554899576359x_{39} = 7.34554899576359
x40=−17.6943623327888+30.2268490547442ix_{40} = -17.6943623327888 + 30.2268490547442 i
x41=5.87298334620742+7.87298334620742ix_{41} = 5.87298334620742 + 7.87298334620742 i
x42=−23.9595998233815+23.1929481687112ix_{42} = -23.9595998233815 + 23.1929481687112 i
x43=10.2068191433355x_{43} = 10.2068191433355
x44=−17.2367532909385+29.4399675077197ix_{44} = -17.2367532909385 + 29.4399675077197 i
x45=−12.6470126043852−28.118562365869ix_{45} = -12.6470126043852 - 28.118562365869 i
x46=−0.123179693577312−13.8907413537161ix_{46} = -0.123179693577312 - 13.8907413537161 i
x47=−79.6744527677435−40.5724722990315ix_{47} = -79.6744527677435 - 40.5724722990315 i
x48=−601.174648738531+120.78735664152ix_{48} = -601.174648738531 + 120.78735664152 i
x49=−4.35400726624713−22.1197535740408ix_{49} = -4.35400726624713 - 22.1197535740408 i
x50=−24.887423745308+26.6390194257314ix_{50} = -24.887423745308 + 26.6390194257314 i
x51=−8.90583145320022−25.3827892249992ix_{51} = -8.90583145320022 - 25.3827892249992 i
x52=−43.0153722534256+35.0346814665705ix_{52} = -43.0153722534256 + 35.0346814665705 i
x53=8.80293710149006+3.72070624957389ix_{53} = 8.80293710149006 + 3.72070624957389 i
x54=6x_{54} = 6
x55=6.63749646475538−0.515313189443995ix_{55} = 6.63749646475538 - 0.515313189443995 i
x56=10.9672909709961x_{56} = 10.9672909709961
x57=−15.7889029954648+21.431585840444ix_{57} = -15.7889029954648 + 21.431585840444 i
x58=−73.8374983000384−48.4321838201617ix_{58} = -73.8374983000384 - 48.4321838201617 i
x59=−6.15890411996741−23.4013169525299ix_{59} = -6.15890411996741 - 23.4013169525299 i
x60=−31.3073886486869+31.2443284997006ix_{60} = -31.3073886486869 + 31.2443284997006 i
x61=6.17452091729384−4.6765271733698ix_{61} = 6.17452091729384 - 4.6765271733698 i
x62=10.7679491924311x_{62} = 10.7679491924311
x63=10.4252769214024x_{63} = 10.4252769214024
x64=−7.98468079041038−24.7150053322795ix_{64} = -7.98468079041038 - 24.7150053322795 i
x65=−16.0989958917086+20.8268978164982ix_{65} = -16.0989958917086 + 20.8268978164982 i
x66=−16.0148131573779+23.4060300922983ix_{66} = -16.0148131573779 + 23.4060300922983 i
x67=−13.9420887749843+25.7270721438083ix_{67} = -13.9420887749843 + 25.7270721438083 i
x68=6.8237448862837−1.41865451731764ix_{68} = 6.8237448862837 - 1.41865451731764 i
x69=−69.5261729641969+52.9052285159186ix_{69} = -69.5261729641969 + 52.9052285159186 i
x70=6.00122352608875+0.0220373775398462ix_{70} = 6.00122352608875 + 0.0220373775398462 i
x71=−597.009614284563+131.769748792391ix_{71} = -597.009614284563 + 131.769748792391 i
x72=6.54278373449149x_{72} = 6.54278373449149
x73=−611.475714808815+105.797248405971ix_{73} = -611.475714808815 + 105.797248405971 i
x74=8.79179606750063x_{74} = 8.79179606750063
x75=−2.56746456391166−20.8746094911533ix_{75} = -2.56746456391166 - 20.8746094911533 i
x76=−119.826864335618+46.3630799400323ix_{76} = -119.826864335618 + 46.3630799400323 i
x77=6.35219119231979−1.86535239620488ix_{77} = 6.35219119231979 - 1.86535239620488 i
x78=−609.283696673399+107.929896555719ix_{78} = -609.283696673399 + 107.929896555719 i
x79=−599.586888465186+124.421399364341ix_{79} = -599.586888465186 + 124.421399364341 i
x80=−0.794675676050093−19.6713317824368ix_{80} = -0.794675676050093 - 19.6713317824368 i
x81=−121.422256086859+45.5503634176283ix_{81} = -121.422256086859 + 45.5503634176283 i
x82=7.87367006223537+9.69481100887907ix_{82} = 7.87367006223537 + 9.69481100887907 i
x83=−90.7100876432681+59.6326551680948ix_{83} = -90.7100876432681 + 59.6326551680948 i
x84=9.96887112585073x_{84} = 9.96887112585073
x85=−16.7246877670263+26.9696594255178ix_{85} = -16.7246877670263 + 26.9696594255178 i
x86=−465.104655113882+90.6577909146098ix_{86} = -465.104655113882 + 90.6577909146098 i
x87=8.06804976116508−2.06339030301275ix_{87} = 8.06804976116508 - 2.06339030301275 i
x88=−122.213962313452+54.9523487319941ix_{88} = -122.213962313452 + 54.9523487319941 i
x89=−51.6039374038479+46.3643053708792ix_{89} = -51.6039374038479 + 46.3643053708792 i
x90=−13.5964719266668−28.8171828880408ix_{90} = -13.5964719266668 - 28.8171828880408 i
x91=−122.42751394833+46.3285609568429ix_{91} = -122.42751394833 + 46.3285609568429 i
x92=−29.5502504793758+28.1901982491997ix_{92} = -29.5502504793758 + 28.1901982491997 i
x93=−10.7650635017165−26.738379746536ix_{93} = -10.7650635017165 - 26.738379746536 i
x94=0.972504042020286−18.5170006729628ix_{94} = 0.972504042020286 - 18.5170006729628 i
x95=−1.67975311440071−20.2673402106044ix_{95} = -1.67975311440071 - 20.2673402106044 i
x96=−121.10928710798+49.8720700424126ix_{96} = -121.10928710798 + 49.8720700424126 i
x97=8.86137660652729−2.07433915001509ix_{97} = 8.86137660652729 - 2.07433915001509 i
Descartamos las soluciones complejas:
x1=9.41935871894229x_{1} = 9.41935871894229
x2=8.25x_{2} = 8.25
x3=8.09757208771761x_{3} = 8.09757208771761
x4=7.34554899576359x_{4} = 7.34554899576359
x5=10.2068191433355x_{5} = 10.2068191433355
x6=6x_{6} = 6
x7=10.9672909709961x_{7} = 10.9672909709961
x8=10.7679491924311x_{8} = 10.7679491924311
x9=10.4252769214024x_{9} = 10.4252769214024
x10=6.54278373449149x_{10} = 6.54278373449149
x11=8.79179606750063x_{11} = 8.79179606750063
x12=9.96887112585073x_{12} = 9.96887112585073
Las raíces dadas
x6=6x_{6} = 6
x10=6.54278373449149x_{10} = 6.54278373449149
x4=7.34554899576359x_{4} = 7.34554899576359
x3=8.09757208771761x_{3} = 8.09757208771761
x2=8.25x_{2} = 8.25
x11=8.79179606750063x_{11} = 8.79179606750063
x1=9.41935871894229x_{1} = 9.41935871894229
x12=9.96887112585073x_{12} = 9.96887112585073
x5=10.2068191433355x_{5} = 10.2068191433355
x9=10.4252769214024x_{9} = 10.4252769214024
x8=10.7679491924311x_{8} = 10.7679491924311
x7=10.9672909709961x_{7} = 10.9672909709961
son puntos de cambio del signo de desigualdad en las soluciones.
Primero definámonos con el signo hasta el punto extremo izquierdo:
x0≤x6x_{0} \leq x_{6}
Consideremos, por ejemplo, el punto
x0=x6−110x_{0} = x_{6} - \frac{1}{10}
=
−110+6- \frac{1}{10} + 6
=
5.95.9
lo sustituimos en la expresión
−6x−2+(x+7)+−4x−2+(x+2)≥1\sqrt{- 6 \sqrt{x - 2} + \left(x + 7\right)} + \sqrt{- 4 \sqrt{x - 2} + \left(x + 2\right)} \geq 1
−4−2+5.9+(2+5.9)+−6−2+5.9+(5.9+7)≥1\sqrt{- 4 \sqrt{-2 + 5.9} + \left(2 + 5.9\right)} + \sqrt{- 6 \sqrt{-2 + 5.9} + \left(5.9 + 7\right)} \geq 1
1.05031646837369 >= 1

significa que una de las soluciones de nuestra ecuación será con:
x≤6x \leq 6
 _____           _____           _____           _____           _____           _____           _____          
      \         /     \         /     \         /     \         /     \         /     \         /
-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------•-------
       x6      x10      x4      x3      x2      x11      x1      x12      x5      x9      x8      x7

Recibiremos otras soluciones de la desigualdad pasando al polo siguiente etc.
etc.
Respuesta:
x≤6x \leq 6
x≥6.54278373449149∧x≤7.34554899576359x \geq 6.54278373449149 \wedge x \leq 7.34554899576359
x≥8.09757208771761∧x≤8.25x \geq 8.09757208771761 \wedge x \leq 8.25
x≥8.79179606750063∧x≤9.41935871894229x \geq 8.79179606750063 \wedge x \leq 9.41935871894229
x≥9.96887112585073∧x≤10.2068191433355x \geq 9.96887112585073 \wedge x \leq 10.2068191433355
x≥10.4252769214024∧x≤10.7679491924311x \geq 10.4252769214024 \wedge x \leq 10.7679491924311
x≥10.9672909709961x \geq 10.9672909709961
Solución de la desigualdad en el gráfico
02468-8-6-4-2-1010010
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