Question 1092649
.
See the graph below.


The folk wisdom says:   It is better to see once,  than to hear  100  times.



{{{graph( 330, 330, -5.5, 5.5, -5.5, 5.5,
          abs(x), x-1
)}}}


Plots y = |x| (red)  and  y = x-1 (green)



The <U>answer</U> to the first your inequality is:  the solution is the set of ALL real numbers.



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<U>Solution to the first inequality</U>


|x| > x - 1.


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We must re-write this inequality separately for x >= 0  and for  x < 0,  where the expressions for |x| are different.


1.  For x >= 0  you have  |x| = x.  Then the original inequality takes the form 

    x > x - 1.     (1)

    This inequality is true for all values of x. 

    Thus in the domain  x >= 0  the solution for (1), and, hence, for the original inequality is the set  [0,infinity).



2.  For  x < 0 you have  |x| = -x.  Then the original inequality takes the form 

    -x > x - 1.     (2)

     or, equivalently,

     2x < 1.     ( I hope you know elementary operations with inequalities, so
                   such transformations should be clear to you . . . )

     It implies  x < 1/2,  which ALWAYS is true in the domain  x < 0.

     So, in the domain  x < 0 the solution set for (2), and, hence, for the original inequality is the set  (-infinity,0).


3.  <U>Conclusion</U>.  The solution set for the original equation is the entire number line.
</pre>