MODERATE EXERCISES

(3) Solve the following absolute value inequalities. Also, graph the solution on the number line.

 (a) 5  |x + 1|

 (b) |4x + 20|  12

 (c) 

 (d) |x2 - 2x|  0 

 (a) 5  |x + 1|

     5  x + 1   or    -5  x + 1
     x  4             x  -6


     __________________ _ _ _ _ _ _ _ _ _ _ ____________________
                     -6                      4


 (b) |4x + 20|  12

     4x + 20  12   or   4x + 20  -12

     4x  -8             4x  -32

     x  -2              x  -8

    _ _ _ _ _ _____________________ _ _ _ _ _
              -8                  -2


 (c) 

     1/3|x + 1| < 4/5
     |x + 1| < 12/5

     x + 1 < 12/5   or    x + 1 > -12/5
     x < 7/5              x > -17/5

    _ _ _ _ _ _o________________________o_ _ _ _ _ _
            -17/5                      7/5

 (d) |x2 - 2x| > 0
     
     Since absolute values are ALWAYS greater than or equal to 0, we can find out where x2 - 2x = 0
and eliminate those points.

     x2 - 2x = 0
     x(x-2) = 0
     x = 0   or   x-2 = 0
                  x = 2

     Therefore the answer is (-,0)  (0,2)  (2,)

     _______________o__________o_________________
                    0          2