Absolute Value Inequalities

• Geometric Definition:  |x - a| = b  means  x  is b units away from a on the number line.

 
• Algebraic Definition: 

 

 
 
 
 

Example:  Solve   |x + 1| < |2x - 3| - 4

1. Solve each absolute value for its zeros:

    a.  2x – 3 = 0                b.  x + 1 = 0
               x = 3/2                        x = -1

2. Mark them on a number line:

________________|________________________|_____________________
                              -1                                             3/2

3. Work with each interval:

       a.  x < -1                        b.  -1 < x < 3/2                              c.  x > 3/2
           2x – 3 is negative         2x – 3 is negative                        2x – 3 is positive
            x + 1 is negative           x + 1 is positive                             x + 1 is positive

            so the sentences to solve are:
________________|________________________|_____________________
                              -1                                             3/2

-x - 1 < -2x + 3 – 4          x + 1 < -2x + 3 – 4                      x + 1 < 2x - 3 – 4
             x < 0                             3x < -2                                         -x < -8
                                                    x < -2/3                                        x > 8
  In this interval, x < -1,    In this interval, -1 < x < 3/2     In this interval, x > 3/2,
   so the solutions              so –1 < x < -2/3 are the            so x > 8 are the solutions.
   are x < -1.                      solutions.

4. Putting all the solutions together yields  x < -2/3 or x > 8
    (Keep in mind that it is possible for the solutions to be empty set or all reals.)
 

Go to Precalculus Page
Go to Precalculus Lessons Page
Go to Linear Relations and Functions Lessons Page