(EN) - Equação Modular

por Betoneira de Natal Qua 25 maio 2022, 20:19

(EN) A equação

(EN) - Equação Modular Svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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:
a) não possui solução para a < -2
b) possui duas soluções para a > 2
c) possui solução única para a < 2/3
d) possui solução única para -2 < a < 2/3
e) possui duas soluções para -2 < a < 2/3

Resposta: E

Fiz da seguinte maneira:

->

.
1) 2x + 3 = + (ax + 1)
      x(2 - a) = -2
      x = 2/(a-2)

2) 2x + 3 = -ax - 1
      x(a + 2) = -4
      x = (-4) / (a + 2)

Só que não consigo achar gabarito dessa maneira...
Há outro modo q devo usar?

por **Giovana Martins** Qua 25 maio 2022, 20:33

O jeito mais fácil, a meu ver, seria plotando os gráficos de f(x) = |2x + 3| e g(x) = ax + 1.

Ao fazer isso, é fácil ver que: se a < 0 a solução possuirá solução para todo x real, o que elimina a alternativa "a".

Se a = 2, naturalmente, g(x) nunca intersectará f(x), tendo em vista que g(x) será paralela a um dos "ramos" de f(x).

Se a > 2, f(x) e g(x) intersectam-se em apenas um ponto, o que elimina a alternativa "b".

Se a = 0, f(x) e g(x) intersectam-se em dois pontos, logo, duas soluções, o que elimina as alternativas "c" e "d", pois dentro do intervalo -2 < a < 2/3 há, obrigatoriamente, duas soluções já que para a = 0 já temos duas soluções.

por **Giovana Martins** Qua 25 maio 2022, 20:38

Se você souber mexer no Geogebra, crie um controle deslizante (slider se você estiver usando a versão em inglês) em torno do parâmetro "a" para que fique mais fácil a visualização do que eu disse na postagem acima. Se houver dúvidas sobre como fazer isso, me avise.

por Betoneira de Natal Qua 25 maio 2022, 21:05

Ahhhh saquei.
Putz, n tive essa visão de montar a equação, foi bem mais fácil de visualizar

Consegui montar aqui no GeoGebra.
Vlw Giovana! Very Happy