Question

Determinați numărul natural n, astfel încât:
$\frac{n + 1}{3 \times n + 5} < \frac{2}{7}$
​

Answer 1

Răspuns:

0; 1; 2

Explicație pas cu pas:

$\frac{n + 1}{3 \cdot n + 5} < \frac{2}{7} \iff \frac{n + 1}{3 \cdot n + 5} - \frac{2}{7} < 0$

$\frac{7(n + 1) - 2(3n + 5)}{7(3n + 5)} < 0 \\ \frac{7n + 7 - 6n - 10}{7(3n + 5)} < 0 \iff \frac{n - 3}{7(3n + 5)} < 0$

$n - 3 = 0 \implies n = 3$

$7(3n + 5) = 0 \iff 3n + 5 = 0 \implies n = - \frac{5}{3}$

$- \frac{5}{3} < 3$

pentru:

$n \in \Big(- \infty ; - \frac{5}{3} \Big) \implies \frac{n - 3}{7(3n + 5)} > 0$

pentru:

$n \in \Big(- \frac{5}{3};3\Big) \implies \frac{n - 3}{7(3n + 5)} < 0$

pentru:

$n \in \Big(3;+ \infty \Big) \implies \frac{n - 3}{7(3n + 5)} > 0$

.

$\implies n \in \Big[0;3\Big) \cap \mathbb{N} \implies \bf n \in \{0;1;2 \}$