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.
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