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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.
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, ( , ).
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.