Question 972477: If x is less than or equal to 7, then the absolute value of the expression x-7 = ?
The answer is -x+7. I solved it by removing the absolute value bars, which gave me x-7, and -(x-7) = -x+7.
But, I don't understand what this means. Can anyone explain this to me? What does it mean and why?
Thank you for any help anyone can give.
Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website! If x is less than or equal to 7, then the absolute value of the expression x-7 = ?
 ,
Allows x to be assigned as any of 7 or less than 7.
Try a step:
 ,
again, for any x less than 7, this expression on the left will be less than zero, and this is expected as first given condition, "x is less than or equal to 7...".
Next part:
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Already established, is that ; which means, x-7 WILL BE NEGATIVE, or in other words, x-7<0, or may be zero.
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Thinking carefully of the meaning of absolute value, for ,  ;
Again, you need to think through the meaning of absolute value; and drawing this graphed on a number line will help, if needed.
Try this thought:
Let p be a positive real number.
 , nothing too complicated. p is p units away from zero on the number line, and because p was given as POSITIVE, p is p units TO THE RIGHT OF ZERO.
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 is also p units AWAY FROM 0 on the number line but is to the left of zero.
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 -----------What about this?
 , because is negative.
 .
p is a positive number, and -p is a negative number.
Try examening on your own this one:
Let r be a negative real number, or let r be real number such that  .
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What is  ?
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