Question

Resolva em R,as seguintes inequações modulares :
a)|x+3| > 7
b)|5x-3| < 12

Answer 1

a)
X + 3 > 7
x > 7-3
x > 4

b)
5x -3 < 12
5x < 12 + 3
5x < 15
x < 15/5
x < 3

Answer 2

De modo geral, sendo k um numero positivo temos que:

$|x| < k \implies -k<x<k\\\\|x| > k \implies x<-k \:\:ou\:\:x>k$

a)

$|x+3| > 7 \implies \left \{ {{x+3<-7} \atop {x+3>7}} \right. \\\\x+3<-7\implies x+3-3<-7-3\implies x<-10\\\\x+3>7\implies x+3-3>7-3\implies x>4\\\\S= \{x\in \mathbb{R}|x<-10\:ou\:x>4\}$

b)

$|5x-3| < 12 \implies -12<5x-3 < 12 \\\\-12<5x-3 < 12\implies -12+3<5x-3+3 < 12+3\implies -9<5x < 15\\\implies -\dfrac{9}{5}<x<3 \\\\S= \{x\in \mathbb{R}|-\dfrac{9}{5}<x<3 \}$