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In my class, it was determined that when an absolute value inequality has a negative answer, then it is no solution. However I want to know what the difference is and if there is any way to determine whether it is all real numbers or no solution with a glance or if theres any theories behind it... 

Take these two examples: 

\(3+2|9+x|≤ -1 \)     This is supposed to have no solutions 


\(-1+4|6x|>-97\)  This is supposed to be all real numbers 

They both have a negative, and we know that an absolute value cannot be negative. If anyone can clarify this and tell me the difference between the no solution and all real numbers i would really appreciate it because I have an exam tomorrow and I REALLY need clarification on this. 

Thank you! 

 Aug 30, 2019

3 + 2l 9 + xl  ≤  -1      subtract  3 from both sides


2 l 9 + x l  ≤  -4          divide both sides by 2


l 9 + x l  ≤   -2 


Since the result of an absolte value is either  positive or zero, then the result of the absolute value on the left cannot be ≤ -2



-1 + 4 l 6x l   >  -97     add 1 to both sides


4 l 6x l >  -96                divide both sides by 4


l 6x l  >   -24


For the same reason as the first......the minimum value of  the absolute value for any real number x will  = 0

And 0  > -24    is always true



cool cool cool

 Aug 30, 2019

Ahh that makes more sense!!  Thank you so much :)) 

Nirvana  Sep 1, 2019

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