Absolute Value Equations Equations Involving Absolute Values To solve an equation that involves an absolute value, turn the equation into an "Or" statement. Example  Solve          |x|  =  2  Solution         x  =  2 or x  =  -2   Example          |x + 1|  =  2 Solution         x + 1  =  2    or    x + 1  =  -2         x  =  1     or     x  =  -3                  Subtracting 1 from both sides Example         |3x - 4|  =  5  Solution We write         3x - 4  =  5     or     3x - 4  =  -5         3x  =  9     or     3x  =  -1                    Adding 4 to both sides         x  =  3     or     x  =  -1/3                     Dividing by 3 Example         | 4x + 7|  =  -3   Solution This equation has no solution, since an absolute value cannot be negative.   Example        | 2x - 6|  =  0   Solution Since positive and negative 0 mean the same thing, we only need one equation         2x - 6  =  0         2x  =  6         x  =  3     Exercises Solve the following.  Put your mouse on the yellow rectangle to check your solution. |x|  =  6 |x - 3|  =  6 |x|  =  -4 |2x + 1|  =  7 Equations That have Absolute Value Signs on Both Sides If we have absolute value signs on both sides of the equation, we can play the same game with two choices as follows. Example:   Solve          |3x + 4|  =  | 2x - 3| Solution: We can write         3x + 4  =  2x - 3      or     3x + 4  =  -(2x - 3)         3x + 4  =  2x - 3     or     3x + 4  =  -2x + 3          Distributing the (-)         3x  =  2x - 7     or     3x  =  -2x - 1                        Subtracting 4         x  =  -7      or     5x  =  -1                                      Subtracting (Adding)  the 2x          x  =  -7     or      or x   =  -1/5                                    Dividing by 5   Back to Math 152A Home Page Back to the Math Department Home e-mail Questions and Suggestions