Absolute Value


Absolute Value


Absolute value is a number's distance from 0.

     |6|  =  6

     |−8|  =  8

     |0|  =  0

When solving equations or inequalities involving absolute value, it may be helpful to translate the expression into English in order to understand what the expression means. 


|x| = 5 states that x is exactly 5 units away from zero.

  • This expression may be written as x = 5 or x = −5.
  • The solution may be expressed with a set:  {−5, 5}.

|x| < 5  states that x is less than 5 units away from zero. 

  • Either x is greater than −5 or less than +5.
  • This expression may be written −5 < x < 5.
  • The solution may be expressed with an interval:  (−5, 5).

|x| > 5  states that x is more than 5 units away from zero. 

  • Either x is greater than 5 or x is less than −5.  
  • This expression may be written x > 5 or x < −5.
  • The solution may be expressed with a union of intervals:  (−, −5) ⋃ (5, ).



EXAMPLE 1

Solve the equation |3x − 5| = 13.


You can re-express the equation as 3x − 5 = 13 or 3x − 5 = 13.

Solving these equations separately:

     3x − 5 + 5 = 13 + 5          3x − 5 + 513 + 5

     3x = 18                            3x8

     3x/3 = 18/3                     3x/3 = 8/3

     x = 6                                x8/3

The solution to the equation is the set x = {8/3, 6}.


EXAMPLE 2

Solve and graph the inequality |2x + 3| < 9.


Re-expressing the inequality as 9 < 2x + 3 < 9, we can solve this by operating on all three sides:

     9 − 3 < 2x + 3 − 3 < 9 − 3

     12 < 2x < 6

     12/2 < 2x/2 < 6/2

     6 < x < 3

The solution is the interval (6, 3), and is graphed on the number line as .


Remember... whenever you divide an inequality by a negative number, the symbol(s) change direction.




Practice

Solve the following equations.

1.  |x + 4| = 12


2.  |7 − 2x| = 15


3.  |5x + 1| − 3 = 16


Solve the following inequalities.  Express your answers as intervals.

4.  |3x − 2| ≤ 7


5.  |6x + 5| > 9


6.  |4x + 3| < −5















Answers

1.  {−16, 8}

2.  {−4, 11}

3.  {−4, }

4.  [, 3]

5.  (−, ) ⋃ (, )

6.  ⊘