ALGEBRA - solved problems
  Linear equations in one variable
Absolute value equations
Solving absolute value equations
79.        | 1 - 2x | = 17
Solution:        1 - 2x = 17                             or                      1 - 2 x = -17
                           2 x = - 16                                                         2x = 18
                             x = - 8                                                             x = 9
The solutions to the given equation are x = - 8 and x = 9.
80.          | -3 - x | = 5
Solution:          -3 - x = 5                             or                      -3 - x = -
                              x = - 8                                                            x = 2
The solutions to the given equation are x = - 8 and x = 2.
81.         | x + 1 | = 2x - 3
Solution:        x + 1 = 2x - 3                             or                      x + 1 = -(2 x - 3)
                     x - 2x = - 4                                                         x + 2x = 3 - 1
                             x = 4                                                                   3x = 2     =>    x = 2/3
Check solutions:
   x = 4   =>     | x + 1 | = 2x - 3,                                x = 2/3    =>    | x + 1 | = 2x - 3 
                        | 4 + 1 | = 2 4 - 3                                                   | 2/3 + 1 | = 2 2/3 - 3
                                  5 = 5                                                                         5/3 is not -5/3
The check shows that x = 2/3 is not a solution, because the right side of the equation becomes negative. There is a single solution to this equation: x = 4.
82.        | x + 2 | = | 2x - 5 |
Solution:    As both sides of the equation contain absolute values the only way the two sides are equal is,
the two quantities inside the absolute value bars are equal or equal but with opposite signs.
                      x + 2 = 2x - 5                             or                      x + 2 = -(2 x - 5)
                     x - 2x = - 5 - 2                                                     x + 2 = -2 x + 5
                            -x = -7                                                                 3x = 3     
                              x = 7                                                                    x = 1
Check solutions:
   x = 7    =>    | x + 2 | = | 2x - 5 |,                             x = 1   =>     | x + 2 | = | 2x - 5 | 
                        | 7 + 2 | = | 2 7 - 5 |                                               | 1 + 2 | = | 2 1 - 5 |
                                  9 = 9                                                                        3 = | -3 |
Therefore, the solutions to the given equation are  x = 7 and  x = 1.
Linear equation with absolute value, graphic solution
83.          | -3 - x | = 2.
Solution:
-3 - x = 2          or           -3 - x = -2
                x = - 5                             x = -1
The solutions to the given equation are
x = - and  x = -1.
84.        | x + 1 | = 2x - 3.
Solution:       x + 1 = 2x - 3                             or                      x + 1 = -(2 x - 3)
                     x - 2x = - 4                                                         x + 2x = 3 - 1
                             x = 4                                                                   3x = 2    =>    x = 2/3
Check solutions:
   x = 4   =>     | x + 1 | = 2x - 3,                                x = 2/3   =>     | x + 1 | = 2x - 3 
                        | 4 + 1 | = 2 4 - 3                                                   | 2/3 + 1 | = 2 2/3 - 3
                                  5 = 5                                                                         5/3  is not  -5/3
The check shows that x = 2/3 is not a solution, because the right side of the equation becomes negative. There is a single solution to this equation: x = 4.
85.          | x + 2 | = | 2x - 5 |.
Solution:   As both sides of the equation contain absolute values the only way the two sides are equal is, the two quantities inside the absolute value bars are equal or equal but with opposite signs.
                      x + 2 = 2x - 5                             or                      x + 2 = -(2 x - 5)
                     x - 2x = - 5 - 2                                                     x + 2 = -2 x + 5
                            -x = -7                                                                 3x = 3     
                              x = 7                                                                    x = 1
Check solutions:
   x = 7   =>     | x + 2 | = | 2x - 5 |,                             x = 1   =>     | x + 2 | = | 2x - 5 | 
                        | 7 + 2 | = | 2 7 - 5 |                                               | 1 + 2 | = | 2 1 - 5 |
                                  9 = 9                                                                        3 = | -3 |
Therefore, the solutions to the given equation are  x = 1 and  x = 7.
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