Question

Ajutatima va rog dau 80 puncte!!!!!!
Rezolvati in R ecuatiile:
a) |3x-1| = |1-x|
b) l2+xl = l5x-3l
c)2lxl = lx-3l+2

Answer 1

a)

|3x-1| = |1-x|

3x-1=1-x
3x-1=-(1-x)

3x+x=1+1
3x-1=-1+x

4x=2
3x-x=-1+1

x=2:4
2x=0

x=1/2
x=0

x∈{0, 1/2}

b)

|2+x| = |5x-3|

2+x=5x-3
2+x=-(5x-3)

x-5x=-3-2
2+x=-5x+3

-4x=-5
x+5x=3-2

x=-5:-4
6x=1

x=5/4
x=1/6

x∈{1/6, 5/4}

c)

2|x| = |x-3|+2 ⇔ x≠0

Aici von avea 4 situatii:

x-3=0
x-3<0, x>0
x-3<0, x<0
x-3>0

Daca x-3=0
          x=3

2×3=0+2
6=2   (fals)

Daca x-3>0
         x>3 (x este pozitiv)

2x=x-3+2
2x-x=-3+2
x=-3 (fals)

Daca x-3<0, x>0

2x=-(x-3)+2
2x=-x+3+2
2x+x=5
3x=5
x=5/3

Daca x-3<0, x<0

2(-x)=-(x-3)+2
-2x=-x+3+2
-2x+x=3+2
-x=5
x=-5

x∈{5/3, -5}