SOLUTION: Solve the inequality|x-2|>2x-5

Algebra ->  Permutations -> SOLUTION: Solve the inequality|x-2|>2x-5      Log On


   



Question 1159458: Solve the inequality|x-2|>2x-5
Found 2 solutions by MowMow, ikleyn:
Answer by MowMow(42) About Me  (Show Source):
You can put this solution on YOUR website!
|x-2|>2x-5
a. x-2>2x-5
3>x or x<3
and
b. x-2 < 5-2x
3x<7
x<7/3

Answer by ikleyn(52777) About Me  (Show Source):
You can put this solution on YOUR website!
.
Solve the inequality |x-2| > 2x-5.
~~~~~~~~~~~~~~~~~~~~~


        The solution by  @MowMow in his post is  FATALLY  wrong.
        I came to bring a correct solution.


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.


ANSWER.  The solution set is  x < 3,  or, in interval notation,  ( -infinity,3 ).

Solved.

--------------------------

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.