SOLUTION: Solve the inequalities for the range of x if any. |x|+|2-x|<2

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Question 1192806: Solve the inequalities for the range of x if any.
|x|+|2-x|<2
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!



Use the definition of absolute value 



Isolate the more complicated square root on the left:



Square both sides:









Divide both sides by -4 which flips the < to >:



This can never be true because if x were positive,
equality would hold.   If x were 0, equality would
hold, and if x were negative, < would hold.

So the inequality is a contradiction.

Edwin
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The other tutor uses a method that I would not choose; there are two other standard solution methods that I think are easier.

And his final statement that the inequality is a contradiction is not mathematically correct; his statement should be that the solution set to the inequality is the empty set.

First alternative method....

|x-a| can be interpreted as the distance between x and a. So

|x| = |x-0|
|2-x| = |x-2|

Then the inequality can be read as "the sum of the distances from x to 0 and from x to 2 is less than 2". Picturing points on a number line, it is easy to see that |x|+|2-x| is EQUAL to 2 for 0 < x < 2; and |x|+|2-x| is greater than 2 everywhere else. So there is no solution the inequality; its solution is the empty set.

Second alternative method....

The terms |x| and |2-x| mean the behavior of the expression changes at x=0 and x=2. So analyze the three cases for x less than 0, for x between 0 and 2, and for x greater than 2.

(1) If x is less than 0, then |x| = -x and |2-x| = 2-x. The inequality is then
(-x)+(2-x)<2
-2x+2<2
-2x<0
x>0

But that "solution" is not in the range of x values we are working with (x less than 0). So there is no solution for x less than 0.

(2) If x is between 0 and 2, then |x| = x and |2-x| = 2-x; the inequality is then
(x)+(2-x)<2
2<2

That statement is never true; so there is no solution for x between 0 and 2.

(3) If x is greater than 2, then |x| = x and |2-x| = x-2; the inequality is then
(x)+(x-2)<2
2x-2<2
2x<4
x<2

But that "solution" is again not in the range of x values we are working with (x greater than 2). So there is no solution for x greater than 2.

This shows that there is no solution to the inequality in any of the three intervals; therefore, there is no solution to the inequality -- the solution is the empty set.
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