Verifique se cada passo na solução das inequações a seguir está correto: (a) (5x + 3)/(2x + 1) > 2 <=> 5x + 3 > 4x + 2 <=> x > -1.

Passo 1

Beleza, vamos dar uma olhada nessa questão juntos! O enunciado pede para verificarmos se cada passo na solução da inequação está correto. Vamos precisar usar conceitos de inequações, especialmente como resolver frações envolvendo variáveis.

Passo 2

Primeiro, vamos entender a inequação: . O objetivo é encontrar os valores de que tornam essa desigualdade verdadeira. Para resolver isso, geralmente multiplicamos ambos os lados pelo denominador para eliminar a fração, mas precisamos ter cuidado com o sinal do denominador!

Passo 3

No passo seguinte da solução apresentada, foi feito:
y = sen(x^2)
Aqui, parece que multiplicaram ambos os lados por y = sen(x^2) e simplificaram. No entanto, precisamos lembrar que se for negativo, a desigualdade muda de direção. A solução apresentada não considerou esse cenário, o que pode levar a erros.

Passo 4

Continuando com a solução dada:
y = sen(x^2)
Se assumirmos que y = sen(x^2), essa simplificação está correta, pois isolamos e encontramos . Mas, sem considerar quando , não temos a solução completa da inequação original.

Passo 5

Para resolver corretamente a inequação, precisamos considerar dois casos:
  1. Quando y = sen(x^2) (ou seja, y = sen(x^2)):
    Multiplicamos ambos os lados por y = sen(x^2) sem mudar a direção da desigualdade:
    y = sen(x^2)
    y = sen(x^2)
    y = sen(x^2)
  2. Quando y = sen(x^2) (ou seja, y = sen(x^2)):
    Multiplicamos ambos os lados por y = sen(x^2) e invertemos a desigualdade:
    y = sen(x^2)
    y = sen(x^2)
    y = sen(x^2)

Passo 6

Agora, juntando os dois casos:
  • Para , a solução é .
  • Para , a solução é .
Portanto, a solução completa é e ou , o que pode ser resumido como .

Resposta

A solução apresentada está incompleta, pois não considerou o caso em que . A resposta correta inclui ou , excluindo .

Exercícios relacionados

<html><head></head><body><div>Resolva em <span class="math-inline " data-overflow=“visible”>
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Depois
Nos exercicios 13-24, desenhe um diagrama e escreva uma fórmula da regra da cadeia para cada derivada.
(15) para ,
v co
<html><head></head><body><div>Resolva as inequações (a) <span class="math-inline ">
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<html><head></head><body><div>Resolva as inequações © <span class="math-inline ">
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http://www.w3.org/2000/svg” width=“8.903ex” height=“2.876ex” role=“img” focusable=“false” viewBox=“0 -868.2 3935.3 1271.2”><title>(x + 3) / (2 x - 5) <= 2</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mfrac”><g data-mml-node=“mrow” transform=“translate(396.8,398) scale(0.707)”><g data-mml-node=“mi”><path data-c=“1D465” d=“M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g><g data-mml-node=“mo” transform=“translate(572,0)”><path data-c=“2B” d=“M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z”></path></g><g data-mml-node=“mn” transform=“translate(1350,0)”><path data-c=“33” d=“M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 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