Question

1) Resolva as seguintes equações modulares:
a) | x+3 | = 7
b) |3x-8 | = 13
c) | 3x+2 | = x+1
d) | 3x+1 | = | x-3 |
e) | 6x+4 | = | x-2 |
f) |3x – 1| = |2x + 6|
g) |4x + 3| = – 3x + 7
h) |2x – 1| = x – 1
i) |x2 – x – 2| = 2x + 2

Answer 1

Como está no módulo, basta tirar o sinal dos números e resolver normalmente.

A)
x + 3 = 7
x = 7 - 3
x = 4

B)
3x + 8 = 13
3x = 18 - 3
3x = 15
x = 15/3
x = 5

C)
3x + 2 = x + 1
3x - x = 1 - 2
2x = -1
x = -1/2

D)
3x + 1 = x + 3
3x - x = 3 - 1
2x = 2
x = 2/2
x = 1

E)
6x + 4 = x + 2
5x - x = 2 - 4
4x = -2
x = -2/4
x = -1/2

F)
3x + 1 = 2x + 6
3x - 2x = 6 - 1
x = 5

G)
4x + 3 = -3 + 7
4x = -3 - 3 + 7
4x = -6 + 7
4x = 1
x = 1/4

H)
2x + 1 = x - 1
2x - x = - 1 - 1
x = -2

I)
x^2 + x + 2 = 2x + 2
x^2 +x -2x +2 -2 = 0
x ^2 -x = 0
Fator comum em evidência:
x (x - 1) = 0
x = 0
ou
x - 1 = 0
x = 1

Answer 2

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

$|x|=k \implies x=k \: ou \:x=-k$

a)

$| x+3 | = 7 \implies \left \{ {{x+3=7} \atop {x+3=-7}} \right. \\\\x+3=7 \implies x=4\\\\x+3=-7 \implies x=-10\\\\S= \{-10, \: 4\}$

b)

$|3x-8 | = 13 \implies \left \{ {{3x-8 = 13} \atop {3x-8 = -13}} \right. \\\\3x-8 = 13\implies3x = 21 \implies x=7\\\\3x-8 = -13\implies3x = -5 \implies x=-\dfrac{5}{3} \\\\S= \{-\dfrac{5}{3}, \: 7\}$

c)

$| 3x+2 | = x+1 \implies \left \{ {{3x+2 = x+1} \atop {3x+2= -x-1}} \right. \\\\3x+2 = x+1\implies3x-x=1-2\implies x=-\dfrac{1}{2} \\\\3x+2 = -x-1\implies3x+x=-1-2\implies x=-\dfrac{3}{4} \\\\S= \{-\dfrac{3}{4}, \: -\dfrac{1}{2}\}$

d)

$| 3x+1 | = | x-3 | \implies \left \{ {{3x+1 = x-3} \atop {3x+1= -x+3}} \right. \\\\3x+1 = x-3\implies3x-x=-3-1\implies x=-2 \\\\3x+1 = -x+3\implies3x+x=3-1\implies x=\dfrac{1}{2} \\\\S= \{-2,\: \dfrac{1}{2}\}$

e)

$| 6x+4 | = | x-2 | \implies \left \{ {{6x+4 = x-2 } \atop {6x+4 = -x+2}} \right. \\\\6x+4 = x-2 \implies6x-x=-2-4\implies x=-\dfrac{6}{5} \\\\6x+4 =-x-2 \implies6x+x=2-4\implies x=-\dfrac{2}{7}\\\\S= \{-\dfrac{6}{5},\:- \dfrac{2}{7}\}$

f)

$|3x - 1| = |2x + 6|\implies \left \{ {{3x - 1 = 2x + 6} \atop {3x -1 = -2x -6}} \right. \\\\3x - 1 = 2x + 6\implies3x-2x=6+1\implies x=7 \\\\3x - 1 = -2x - 6\implies3x+2x=-6+1\implies x=-1\\\\S= \{-1,\:7\}$

g)

$|4x + 3| = -3x + 7\implies \left \{ {{4x + 3 = -3x + 7} \atop { 4x + 3 = 3x -7}} \right. \\\\4x + 3 = -3x + 7\implies4x+3x=7-3\implies x=\dfrac{4}{7} \\\\4x + 3 = 3x - 7\implies4x-3x=-7+3\implies x=-4\\\\S= \{-4,\:\dfrac{4}{7}\}$

h)

$|2x - 1| = x -1\implies \left \{ {{2x - 1 = x -1} \atop { 2x - 1= -x +1} \right. \\\\2x - 1 = x -1\implies2x-x=-1+1\implies x=0 \\\\2x - 1 = -x +1\implies2x+x=1+1\implies x=\dfrac{2}{3} \\\\S= \{0,\:\dfrac{2}{3}\}$

i)

$|x^2 - x - 2| = 2x + 2\implies \left \{ {{x^2 - x - 2 = 2x + 2} \atop {x^2 - x - 2 = -2x- 2} \right. \\\\x^2 - x - 2 = 2x + 2\implies x^2 -3x - 4 = 0\implies x=4 \: ou \: x=-1 \\\\x^2 - x - 2 = -2x -2\implies x^2 +x= 0\implies x=-1 \: ou \: x=0 \\\\S= \{-1,\: 0, \:4\}$