Analisar se cada uma das seguintes afirmações é verdadeira. Caso seja falsa, apresente um contraexemplo:
Passo 1
Beleza, vamos lá! O enunciado pede pra gente analisar se a afirmação
é verdadeira para todos os valores de
e
. Para isso, vamos usar conhecimentos de álgebra, especificamente sobre frações e equações. Vamos precisar manipular a equação para ver se ela sempre se mantém verdadeira ou se encontramos algum caso onde ela não funciona.
Passo 2
Primeiro, vamos tentar resolver a equação para ver se ela é válida para todos os
e
. Vamos começar multiplicando ambos os lados da igualdade por
para eliminar as frações:
Simplificando:
Então:
Passo 3
Agora, vamos reorganizar essa equação para ver se conseguimos encontrar uma relação entre
e
:
Podemos fatorar a expressão adicionando e subtraindo 1:
Passo 4
A equação
nos mostra que a afirmação original só é verdadeira quando o produto
é igual a 1. Isso significa que nem todo par de valores
e
vai satisfazer a igualdade original. Ou seja, a afirmação não é verdadeira para todos os
e
.
Passo 5
Para confirmar que a afirmação é falsa em geral, vamos encontrar um contraexemplo onde a igualdade não se mantém. Vamos escolher
e
:
Substituindo na afirmação original:
Ok, aqui deu certo. Vamos tentar outro conjunto de valores, por exemplo,
e
:
Resposta
A afirmação
não é verdadeira para todos os valores de
e
. Um contraexemplo é
e
, onde a igualdade não se sustenta.
Exercícios relacionados
<html><head></head><body><div>Analisar se cada uma das seguintes afirmações é verdadeira. Caso seja falsa, apresente um contraexemplo: <span class="math-inline "></span></div>
</body></html>
</body></html>
<html><head></head><body><div>Calcular <span class="math-inline ">
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<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -1.599ex;” xmlns=“http://www.w3.org/2000/svg” width=“3.979ex” height=“4.328ex” role=“img” focusable=“false” viewBox=“0 -1206.6 1758.8 1913.1”><title>(1/4 * 1/6) / (1/6)</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(220,595) scale(0.707)”><g data-mml-node=“mfrac”><g data-mml-node=“mn” transform=“translate(220,394) scale(0.707)”><path data-c=“31” d=“M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z”></path></g><g data-mml-node=“mn” transform=“translate(220,-345) scale(0.707)”><path data-c=“34” d=“M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 675Q336 677 355 677H373L379 671V211H471V165H379V114Q379 73 379 66T385 54Q393 47 442 46H471V0H462ZM293 211V545L74 212L183 211H293Z”></path></g><rect width=“553.6” height=“60” x=“120” y=“220”></rect></g><g data-mml-node=“mo” transform=“translate(793.6,0)”><path data-c=“22C5” d=“M78 250Q78 274 95 292T138 310Q162 310 180 294T199 251Q199 226 182 208T139 190T96 207T78 250Z”></path></g><g data-mml-node=“mfrac” transform=“translate(1071.6,0)”><g data-mml-node=“mn” transform=“translate(220,394) scale(0.707)”><path data-c=“31” d=“M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z”></path></g><g data-mml-node=“mn” transform=“translate(220,-345) scale(0.707)”><path data-c=“36” d=“M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z”></path></g><rect width=“553.6” height=“60” x=“120” y=“220”></rect></g></g><g data-mml-node=“mfrac” transform=“translate(598.9,-451.6) scale(0.707)”><g data-mml-node=“mn” transform=“translate(220,394) scale(0.707)”><path data-c=“31” d=“M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z”></path></g><g data-mml-node=“mn” transform=“translate(220,-345) scale(0.707)”><path data-c=“36” d=“M42 313Q42 476 123 571T303 666Q372 666 402 630T432 550Q432 525 418 510T379 495Q356 495 341 509T326 548Q326 592 373 601Q351 623 311 626Q240 626 194 566Q147 500 147 364L148 360Q153 366 156 373Q197 433 263 433H267Q313 433 348 414Q372 400 396 374T435 317Q456 268 456 210V192Q456 169 451 149Q440 90 387 34T253 -22Q225 -22 199 -14T143 16T92 75T56 172T42 313ZM257 397Q227 397 205 380T171 335T154 278T148 216Q148 133 160 97T198 39Q222 21 251 21Q302 21 329 59Q342 77 347 104T352 209Q352 289 347 316T329 361Q302 397 257 397Z”></path></g><rect width=“553.6” height=“60” x=“120” y=“220”></rect></g><rect width=“1518.8” height=“60” x=“120” y=“220”></rect></g></g></g></svg></mjx-container></span></div>
</body></html>
<html><head></head><body><div>Analisar se cada uma das seguintes afirmações é verdadeira. Caso seja falsa, apresente um contraexemplo: <span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.486ex;” xmlns=“http://www.w3.org/2000/svg” width=“16.742ex” height=“2.373ex” role=“img” focusable=“false” viewBox=“0 -833.9 7400 1048.9”><title>a^(-2) = -a^2, a != 0</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“msup”><g data-mml-node=“mi”><path data-c=“1D44E” d=“M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z”></path></g><g data-mml-node=“TeXAtom” 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</body></html>
<html><head></head><body><div>Analisar se cada uma das seguintes afirmações é verdadeira. Caso seja falsa, apresente um contraexemplo: <span class="math-inline "></span></div>
</body></html>
</body></html>