Question

Sa se rezolve ecuatia :
[(2x+1)/3]=[(3x+1)/4]
(parentezele patrate sugereaza partea intreaga a numarului)

Answer 1

$[ \frac{2x+1}{3} ]=[ \frac{3x+1}{4} ]$
tinind cont ca $[ \frac{2x+1}{3} ]=[ \frac{3x+1}{4} ]$
egalitatea poate avea loc doar daca
$[ \frac{2x+1}{3} ]=0 ; [ \frac{3x+1}{4} ]=0$
adica daca  $0 \leq \frac{2x+1}{3}\ \textless \ 1$  si  $0 \leq \frac{3x+1}{4}\ \textless \ 1$
$\frac{2x+1}{3}\ \textless \ 1 \\ 2x+1\textless 3 \\ 2x\textless 3-1 \\ x\textless 2 \\ \frac{2x+1}{3}\ \geq 0 \\ [tex]2x \geq -1 \\ x \geq - \frac{1}{2}$
x∈[-1/2,2)
$[ \frac{3x+1}{4} ]=0 \\ \\ \frac{3x+1}{4} \geq 0 \\ \\ 3x+1 \geq 0 \\ 3x \geq -1 \\ x \geq - \frac{1}3}$
3x+1<4 ⇒3x<3⇒x<3
x∈[-1/3,3)

deci  x∈[-1/2,2) intersectat cu x∈[-1/3,3)