Question

O conjunto solução da equação |2x + 3| = |x + 2|, é

Answer 1

Resposta:

3/2(x+2) = 1/2x-4 - 2/x^2-4. Resposta: conjunto vazio.

Answer 2

Resposta:

$\sf |2x + 3| = |x + 2|$

$| \sf x |= \begin{cases} \sf x, & \mbox{se}\quad \sf x \ge 0 \\ \sf - x, & \mbox{se}\quad \sf x < 0\end{cases}$

$\sf 2x + 3 = (x + 2) \; ( l )$

$\sf 2x + 3 = - (x + 2) \; ( l l)$

Resolvendo a equação I temos:

$\sf 2x + 3 = (x + 2)$

$\sf 2x + 3 = x + 2$

$\sf 2x - x = 2 - 3$

$\boxed { \boxed {\boldsymbol{ \sf \displaystyle x = - 1 }} } \quad \gets$

Resolvendo a equação II temos:

$\sf 2x + 3 = -\;(x + 2)$

$\sf 2x + 3 = -\; x - 2$

$\sf 2x + x = - 2- 3$

$\sf 3x = - 5$

$\boxed { \boxed {\boldsymbol{ \sf \displaystyle x = - \frac{5}{3} }} } \quad \gets$

Verificar:

$\sf \mid 2x +3\mid = \mid x +2 \mid$

$\sf \mid 2\cdot (- 1) +3\mid = \mid -1 +2 \mid$

$\sf \mid -2 +3\mid = \mid 1 \mid$

$\sf \mid -1\mid = \mid 1 \mid$

$\sf 1 = 1 \; \checkmark$

$\sf \mid 2x + 3\mid = \mid x+2 \mid$

$\sf \mid 2\cdot\left ( - \frac{5}{3} \right ) + 3\mid = \mid \left ( \frac{1}{2} \right )+2 \mid$

$\sf \mid - \dfrac{10}{3} +3 \mid = \mid - \dfrac{5}{2} + 2 \mid$

$\sf \mid - \dfrac{1}{3} \mid = \mid \dfrac{1}{3} \mid$

$\sf \dfrac{1}{3} = \dfrac{1}{3} \; \checkmark$

$\boldsymbol{ \sf \displaystyle S = \bigg \{ -1; -\; \dfrac{5}{3} \bigg \} }$

Explicação passo-a-passo: