Question 1159458
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Solve the inequality |x-2| > 2x-5.
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        The solution by  @MowMow in his post is  FATALLY  wrong.

        I came to bring a correct solution.



<pre>
We should consider two cases.


            Case 1.


If x >= 2, then the difference  (x-2)  is positive, and |x-2| = x-2.
So, we should find the solutions to this inequality

    x-2 > 2x - 5     (1)

in the domain  x >= 2.


Inequality  (1)  is equivalent to

    5 - 2 > 2x - x,

or

      3   > x.


Thus, in the domain  x >= 2,  the solution set is  x < 3.

In other words, in the domain  x >= 2  the solution set is  2 <= x < 3.



            Case 2.


If x < 2, then the difference (x-2)  is negative, and |x-2| = -(x-2).
So, we should find the solutions to this inequality

    -(x-2) > 2x - 5    (2)

in the domain x < 2. 


Inequality (2) is equivalent to

    -x + 2 > 2x - 5,

    5  + 2 > 2x + x

      7   >    3x

      x   <    7/3.


Thus in the domain  x < 2, all the values satisfy inequality (2).


Combining cases 1 and 2, we see that the solution set for the original equation is  x < 3.


<U>ANSWER</U>.  The solution set is  x < 3,  or, in interval notation,  ( {{{-infinity}}},{{{3}}} ).
</pre>

Solved.


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In the online picture


https://www.desmos.com/calculator/nzwdgckzfi 


I prepared a plot of functions  y = |x-2|  and  y = 2x-5.


Looking at this plot, you may check visually that my answer is correct.