[absolutevalue equations][section 31]

(1.)  |a + 3| = 2                here is the problem

a + 3 = 2         a + 3 = -2    set = to 2 and to -2

-3  -3             -3  -3    subtract 3 fr ea side
____________      _______________
a = -1  ,      x = -5       subtract

result: a = -1,  a = -5

(2.)  |3x - 6| = 12                here is the problem

3x - 6 = 12         3x - 6 = -12   set = to 12 & to -12

+6  +6             +6   +6    add 6 to each side
_______________      ________________
3x   =  18    ,      3x =   -6        add
___    ___          ____   ____
3      3           3       3     divide ea side by 3

x = 6   ,  x = -2             divide and cancel

result:  x = 6  ,  x = -2

(3.)  |t + 1| = 9               here is the problem

t + 1 = 9     t + 1 = -9   set = to 9 and to -9

-1  -1         -1   -1    subtract 1 from each side
___________   _____________
t = 8   ,     t = -10     subtract

result:  t = 8  ,   t = -10

(4.)   |6k + 6| = 24             here is the problem

6k + 6 = 24        6k + 6 = -24   set equal to 24 & to -24

-6  -6            -6   -6   subtract 6 from each side
_____________    _________________
6k   =  18    ,     6k  =  -30       subtract
___    ___         ___    ______
6        6          6        6       divide ea side by 6

k = 3    ,    k = -5             divide and cancel

result:  k = 3  ,   k = -5

(5.)  |3m + 2| = 17                here is the problem

3m + 2 = 17        3m + 2 = -17    set equal to 17 & to -17

-2   -2            -2   -2     subtract 2 from each side
_____________     _______________
3m  =  15    ,   3m   =   -19       subtract
___   ___        ___      ____
3       3        3          3  divide each side by 3

m = 5   ,     m = -6 1/3        divide and cancel

result:  m = 5   ,    m = -6 1/3

(6.)  |6r + 2| = 10              here is the problem

6r + 2 = 10         6r + 2 = -10   set equal to 10 and to -10

-2  -2            -2   -2    subtract 2 from each side
______________     _______________
6r    = 8         6r   = -12      subtract
__    ___         ___   _____
6      6         6        6      divide each side by 6

r = 1 1/3         r = -2        divide and cancel

result:  r = 1 1/3   ,   r = -2

(7.)   |3(t - 1)| - 4 = 16         here is the problem

|3t - 3| - 4 = 16      multiply thru parentheses

+4   +4       add 4 to each side
__________________
|3t - 3|  =    20              add

3t - 3 = 20       3t - 3 = -20   set equal to 20 and to -20

+   3   +3           +3    +3    add 3 to each side
______________    ______________
3t    = 23   ,   3t  =   -17        add
__     ___      ___      _____
3       3       3        3      divide each side by 3

t = 7 2/3       t = -5 2/3     divide and cancel

result:  t = 7 2/3  ,   t = -5 2/3

(8.)  |7x - 5| = 30              here is the problem

7x - 5 = 30        7x - 5 = -30    set equal to 30 & to -30

+ 5  +5            +5   +5      add 5 to each side
_______________     ____________
7x  = 35     ,    7x =  -25        add
___  ___          __    ____
7    7            7     7       divide each side by 7

x = 5    ,    x = -3 4/7        divide and cancel

result:  x = 5  ,   x = -3 4/7

(9.)   |8 - 2(y - 1)| + 5 = 9        here is the problem

|8 - 2y + 2| + 5 = 9      multiply thru parentheses

|10 - 2y| + 5 = 9      combine like terms

-5  -5        subtract 5 from each side
______________________
|10 - 2y|   =   4         subtract

10 - 2y = 4       10 - 2y = -4    set equal to 4 & to -4

-10 + 2y = -4     -10 + 2y = 4     multiply thru by -1

+ 10    + 10      +10      +10   add 10 to each side
______________     ________________
2y = 6    ,      2y = 14         add
___ ___          ___  ____
2   2             2    2      divide each side by 2

y = 3   ,    y = 7         divide and cancel

result:  y = 3   ,   y = 7

(10.)  12 = |3 - x|             here is the problem

3 - x = 12        3 - x = -12   set equal to 12 and to -12

-3 + x = -12     -3 + x = 12    multiply thru by -1

+  3     +    3    + 3      + 3   add 3 to each side
__________________  _______________
x = -9   ,      x = 15     add

result:  x = -9  ,  x = 15