Question

O conjunto solução da equação
| x-3 | + | x - 2 | = 5 é um subconjunto de:

Answer 1

Vamos identificar as funções nos devidos intervalos e analisar :

$\displaystyle \sf |x-3| = \left\{ \begin{array}{I}\displaystyle \sf x-3 , \text{ se } x\geq 3\\\\ \displaystyle \sf -x+3 , \ \text{se } x < 3 \end{array}$

e

$\displaystyle \sf |x-2| = \left\{ \begin{array}{I}\displaystyle \sf x-2 , \text{ se } x\geq 2\\\\ \displaystyle \sf -x+2 , \ \text{se } x < 2 \end{array}$

$\displaystyle \sf \underline{\text{Para o intervalo } x < 2} : \\\\ -x+2 - x+3 = 5 \\\\ -2x+5 = 5 \\\\ -2x = 0 \\\\ \boxed{\sf x = 0 \ \in \ x < 2} \\\\\\ \underline{\text{Para o intervalo } 2 \leq x < 3} : \\\\ -x+3+x-2 = 5 \\\\ 1 = 5 \ (ABSURDO ) \\\\\\ \underline{\text{Para o intervalo } x\geq 3 } : \\\\ x-3+x-2 = 5 \\\\ 2x -5 = 5 \\\\ 2x= 10 \\\\ \boxed{\sf x = 5\ \in x\geq 3 \ } \\\\\\ \underline{\text{Conjunto Solu{\c c}{\~a}o}} : \\\\ \huge\boxed{\sf S = \{ \ 0\ ,\ 5\ \} }\checkmark$