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# What is the difference between an 'All real numbers' absolute value inequality and a 'No solution' inequality?

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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

and

\(-1+4|6x|>-97\)  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

#1
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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   Aug 30, 2019
#2
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Ahh that makes more sense!!  Thank you so much :))

Nirvana  Sep 1, 2019