Absolute value equations
Solving absolute value equations
Absolute value equations
Solving absolute value equations
Recall that the absolute value of a real number a, denoted | a |, is the number without its sign and represents the distance between 0 (the origin) and that number on the real number line.
Thus, regardless of the value of a number a its absolute value is always either positive or zero, never negative that is,  | a | > 0.
To solve an absolute value equation, isolate the absolute value on one side of the equation, and use the definition of absolute value.
If the number on the other side of the equal sign is positive, we will need to set up two equations to get rid of the absolute value,
- the first equation that set the expression inside the absolute value symbol equal to the other side of the equation,
- and the second equation that set the expression inside the absolute value equal to the opposite of the number on the other side of the equation.
Solve the two equations and verify solutions by plugging the solutions into the original equation.
If the number on the other side of the absolute value equation is negative then the equation has no solution.
Example:     | 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.
Example:     | -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.
Example:     | 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 equal  -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.
Example:     | 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.
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