SOLUTION: The absolute value of {{{ 1- 1/x }}} is less than and equal to 3. Solve the inequality, express your answer in interval notation.

We find the critical values by solving the boundary equation:



Which breaks into these two equations:

  or 

   or 

    or 

   or 

We also have a third critical value x=0 because
that is what we get when we set the denominator
in the original problem = 0.  

So we put those points on a number line

----------------o----o--o------------------
-2       -1   -1/2   0 1/4      1         2

We test a value left of -1/2, say -1, in the
original inequality:






That's true, so we shade the region left of -1/2

<================o----o--o------------------
 -2       -1   -1/2   0 1/4      1         2

We test a value between -1/2 and 0, say -.2 in the
original inequality:








That's false, so we DO NOT shade the region between
-1/2 and 0

We test a value between 0 and 1/4, say .2 in the
original inequality:







That's false, so we DO NOT shade the region between
0 and 1/4.

We test a value right of 1/4, say 1, in the
original inequality:











That's true so we shade -1/2

<================@----o--o=================>
 -2       -1   -1/2   0 1/4      1         2

Now we test the endpoint 1/4







That's true so we shade 1/4

<================@----o--@=================>
 -2       -1   -1/2   0 1/4      1         2

So the solution set is:



Edwin