Question 447772: Can you please help me solve this absolute value inequality?? :)
| 2x + 1 | + | x - 4 | > 0
my calculations are :
|2x + 1| + |x - 4|>0
|2x+1|>-|x-4|
square both sides
4x^2 + 4x + 1 > x^2 - 8x + 16
...3x^2 + 12x - 15 > 0
(3x+15)(x-1)>0
i use quadratic curve and got x<5 , x>1
Found 2 solutions by solver91311, richard1234:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Consider this: |2x + 1| > 0 for all real values of x except x = -1/2. And |x - 4| > 0 for all x except x = 4. So the sum of these two expressions is ALWAYS greater than zero. The inequality is true for all real values of x.
John
My calculator said it, I believe it, that settles it
Answer by richard1234(7193)  (Show Source):
You can put this solution on YOUR website! You are over-thinking the problem, and with that solution, you are likely to make a mistake, since you cannot directly assume you can square both sides with the absolute value signs.
Just note that you have a sum of two absolute value expressions, and the absolute value of any quantity is nonnegative. |2x+1| > 0 unless x = -1/2, and |x-4| > 0 unless x = 4. Since x cannot be -1/2 and 4 at the same time, at least one of the absolute value expressions must be greater than 0, so the inequality holds for all real numbers x.
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