Question

Cine imi rezolva ecuatiile astea sau imi da niste explicatii pt modul gen:
|x-3|≤5 sau |2x+9|≥19 sau |5x-2|>7

Answer 1

1.|x-3|≤5⇒ daca si numai daca   - 5≤ |x-3|≤5

x-3≤5⇒  x≤5+3⇒ x≤8
si
x-3≥-5⇒x≥-5+3⇒x≥-2
x∈apartine intervalului [-2,8]

2.|2x+9|≥19     daca si numai daca    -19≥  | 2x+9| ≥19

2x+9|≥|19
⇒ 2x≥19-9
⇒2x≥10
⇒x≥5
si
2x+9≤-19
⇒2x≤-19-9
⇒2x≤-28
⇒x≤-14
⇒x≤-14
x≥5  si  x≤-14
x∈ (-∞,-14)reunit cu [5,∞]

Answer 2

$a).|x-3| \leq 5 \\ x-3 \leq 5 \\ x \leq 5+3 \\ x \leq 8 \\ x-3 \geq -5 \\ x \geq -5+3 \\ x \geq -2 \\ -2 \leq x \leq 8 \\ x \in [-2,8]$
$\displaystyle b).|2x+9| \geq 19 \\ 2x+9 \geq 19 \\ 2x \geq 19-9 \\ 2x \geq 10 \\ x \geq \frac{10}{2} \\ x \geq 5 \\ 2x+9 \leq -19 \\ 2x \leq -19-9 \\ 2x \leq -28 \\ x \leq - \frac{28}{2} \\ x \leq -14 \\ x \in (- \infty,-14] \cup [5, \infty )$
$\displaystyle c).|5x-2|\ \textgreater \ 7 \\ 5x-2\ \textgreater \ 7 \\ 5x\ \textgreater \ 7+2 \\ 5x\ \textgreater \ 9 \\ x\ \textgreater \ \frac{9}{5} \\ 5x-2\ \textless \ -7 \\ 5x\ \textless \ -7+2 \\ 5x\ \textless \ -5 \\ x\ \textless \ - \frac{5}{5} \\ x\ \textless \ -1 \\ x \in (- \infty ,-1) \cup \left( \frac{9}{5} , \infty \right)$