Question

Rozwiąż wartości bezwzględne .dodatkowo nierówności graficznie

Answer 1

1)
||2x+1|-3|=4
|2x+1|-3 =4    
2x+1-3 =4       2x=6    x=3 
|2x+1|-3=-4     |2x+1| ≠ -1
|2x+1|-3=4
|2x+1|=7
2x + 1 = 7          2x = 6   x = 3
2x + 1 =-7          2x =-8   x = -4
x = {-4; -1; 3}
2)
|5 - 2|x + 3|| = 1
 5-2|x+3|  = 1
-2|x+3|  = -4
|x+3|  = 2
x+3  = 2           x =-1
x+3  =-2           x =-5
5-2|x+3|  =- 1
-2|x+3|  =- 6
|x+3|  = 3
 x+3  = 3          x=0
 x+3  =-3          x=-6
x={-6; -5; -1; 0}
3)
|2|x| -3| = 3
2x-3=3   2x=6   x = 3
2x-3=-3             x = 0
2|x|-3=3
2|x|=6
  |x|=3
   x=3                x = 3
   x=-3               x =-3
x={-3; 0}
4)
||x| + 2| = 10
  x+2 = 10         x = 8  
  x+2 =-10         x =-8
  |x|+2 = 10
  |x| = 8
   x = 8
   x =-8
x = {-8; 8}
5)
||x - 1| -3| = 3
|x-1|-3=3  
x-1-3=3   x = 7
x-1-3=-3  x = 1
|x-1|=6
x-1=6       x = 7
x-1=-6      x =-5
x = {-5; 1; 7}
6)
||x + 2| -4| = 2
|x+2|-4=2
|x+2|=6
x+2=6         x =4
x+2=-6        x=-8
|x+2|-4=-2
|x+2|=2
x+2 =2        x=0
x+2 =-2       x=-4
x = {-8; -4; 0; 4}
7)
||2x + 3| - 2| = 4
2x+3-2=4      2x = 3  x = 1,5
2x+3-2=-4     2x=-5   x =-2,5
|2x+3|-2=4
|2x+3|=6
2x+3=6         2x=3    x = 1,5
2x+3=-6        2x=-9   x =-4,5
x = {-4,5; -2,5; 1,5}
8)
|||2x + 3| -2| -1| = 1
||2x+3|-2|-1=-1
||2x+3|-2|=0
|2x+3|-2 =0
|2x+3| =2
 2x+3 = 2        2x =-1   x  =-0,5
 2x+3 =-2        2x =-5   x  =-2,5
||2x+3|-2|-1=1
||2x+3|-2|=2
|2x+3|-2=2
|2x+3|=4
2x+3=4           2x=1    x = 0,5
2x+3=-4          2x=-7   x =-3,5
|2x+3|-2=-2
|2x+3|=0
2x+3=0           2x=-3   x =-1,5
x={-3,5; -2,5; -1,5; -0,5; 0,5}
9)
|||x + 2| + 4| +1| = 1
||x+2|+4|+1= 1
||x+2|+4|=0
|x+2|+4 =0
|x+2|≠-4
||x+2|+4|+1=-1
||x+2|+4|≠-2
x={∅}
10)
||x|-4|=3
|x|-4=3   
|x|=7            x = 7  x =-7
|x|-4=-3
|x|=1             x = 1  x =-1
x = {-7; -1; 1; 7}
11)
|2 - 3x| < 2
2-3x ≠ 2    x≠0
2-3x≠-2     x≠4/3
|2 - 3x| - 2 < 0
        +         0           _          4/3           +
----------------o---------------------o---------------------------->
x ∈ (0; 4/3)
12)
|5x - 2| ≥ 4
5x-2=4   x = 0,6
5x-2=-4  x =-0,4
|5x - 2| - 4 ≥ 0
        +      -0,4        _       0,6                 +
----------------•------------------•------------------------------------>
x ∈ (-∞; -0,4] ⋃ [0,6; ∞)
13)
||x|-1|≤2
|x|-1=2
|x|=3        x = 3      x =-3
|x|-1=-2
|x|≠-1
||x| - 1| - 2 ≤ 0
           +      -3            _            3              +
-----------------•----------------------•------------------------------------>
x ∈ [-3; 3]
14)
|3 - |x - 2|| ≥ 2
3-|x-2|=2
-|x-2|=-3
|x-2|=3
x-2=3     x = 5
x-2=-3    x =-1
3-|x-2|=-2
-|x-2|=-5
|x-2|=5
x-2=5     x=7
x-2=-5    x=-3
|3 - |x - 2|| - 2 ≥ 0
        +    -3        _     -1        +       5       _       7       +
-------------•---------------•---------------•---------------•-------------------->
x ∈ (-∞; -3] ⋃ [-1; 5] ⋃ [7; +∞)
15)
|2|x| - 1| ≥ 1
2|x|-1=1
2x-1=1
2x=2                  x = 1
2x-1=-1   2x=0   x = 0
|2|x| - 1| - 1| ≥ 0
             +         0           _             1              +
---------------------•----------------------•------------------------------------->
x ∈ (-∞; 0] ⋃  [1; +∞)