Question 1144582
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            I want to show you the short way to solve these problems.



<pre>
(1)  | x - 3 | - 5 >= 2


     It is equivalent to this inequality


     | x - 3 | >= 7.


     The solutions to the last inequality are all numbers in the number line that are remoted from the number 3 to 7 or more units.


     Obviously, these numbers are  x >= 3 + 7 = 10  <U>OR</U>  x <= 3 - 7 = -4.


     So, the solution to the original inequality is the union of two sets  { x <= -4 }  and  { x >= 10 }.    <U>ANSWER</U>



         Notice that <U>all absolute value inequalities OF THIS FORM can be easily solved using this chain of arguments</U>.


         See the lesson
              - <A HREF=https://www.algebra.com/algebra/homework/Inequalities/Solving-absolute-value-inequalities-IK.lesson>Solving absolute value inequalities</A> 
         in this site.



(2)  | 2x + 12 | = -6.


     An absolute value of a number is ALWAYS non-negative.


     It can not be negative.


     So, the given equation HAS NO solutions in real number.   <<<---=== At this point, the solution is just COMPLETED.



      You can make your conclusion <U>even without writing this absolute value equation in any other form</U>.

      Moreover, when you make this conclusion <U>without writing this absolute value equation in any other form</U>,

      you demonstrate that you <U>FIRMLY KNOW</U> what the absolute value is.



      If, in opposite, you try to transform such an equation to any other form, you demonstrate that you are not firm in your knowledge.
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