Question

Resolva as inequações
a: |3x − 1| < 2
b: |3x − 1| < 2
c: |x + 3| > 1
d: |x + 1| < |2x − 1|
e: |x − 3| < x + 1
(f) |x + 1| + |x − 2| >

Answer 1

a)

$\boxed{|3x-1|<2}\ \therefore$

$i)\ 3x-1<2\ \therefore\ 3x<3\ \therefore\ x<1$

$ii)\ 3x-1>-2\ \therefore\ 3x>-1\ \therefore\ x>-\dfrac{1}{3}$

$\boxed{\mathbb{S}=\left\{x\in\mathbb{R}\ |\ -\dfrac{1}{3}<x<1\right\}}$

b)

O mesmo que a resolução anterior.

c)

$\boxed{|x+3|>1}\ \therefore$

$i)\ x+3>1\ \therefore\ x>-2$

$ii)\ x+3<-1\ \therefore\ x<-4$

$\boxed{\mathbb{S}=\left\{x\in\mathbb{R}\ |\ x\in(-\infty,-4[\cup]-2,\infty)\right\}}$

d)

$\boxed{|x+1|<|2x-1|}\ \therefore$

$i)\ x+1=2x-1\ \therefore\ x=2$

$ii)\ x+1=-(2x-1)\ \therefore\ x+1=-2x+1\ \therefore\ x=0$

$x<0\ \to\ \text{p. ex.}\ x=-1\ \to\ 0<3\ \to\ \text{Ok.}\\\\ 0<x<2\ \to\ \text{p. ex.}\ x=1\ \to\ 2\nless1\\\\ x>2\ \to\ \text{p. ex.}\ x=3\ \to\ 4<5\ \to\ \text{Ok.}$

$\boxed{\mathbb{S}=\left\{x\in\mathbb{R}\ |\ x\in(-\infty,0[\cup]2,\infty)\right\}}$

e)

$\boxed{|x-3|<x+1}\ \therefore$

$i)\ x-3=-(x+1)\ \therefore x-3=-x-1\ \therefore\ x=1$

$x<1\ \to\ \text{p. ex.}\ x=0\ \to\ 3\nless1\\\\ x=1\ \to\ 2\nless2\\\\ x>1\ \to\ \text{p. ex.}\ x=2\ \to\ 1<3\ \to\ \text{Ok.}$

$\boxed{\mathbb{S}=\left\{x\in\mathbb{R}\ |\ x\in]1,\infty)\right\}}$

f)

Incompleto. Não é possível resolver.