Rewriting inequality help

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In summary, the inequality states that the absolute value of (x-9) minus the absolute value of x must be greater than or equal to 9. Rewritten, this means that the absolute value of (x-9) must be greater than or equal to the sum of 9 and the absolute value of x. Using the rule for absolute values, this means that x must be less than or equal to 0. However, the solution book may have excluded 0 as a solution, which could potentially be a mistake.
  • #1

danago

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Solve the inequality [tex]
\left| {x - 9} \right| - \left| x \right| \ge 9
[/tex]

I started by rewriting it as:
[tex]
\left| {x - 9} \right| \ge 9 + \left| x \right|
[/tex]

Now, for any real numbers x and y,
[tex]
\left| {x + y} \right| \le \left| x \right| + \left| y \right|
[/tex]

According to that,
|x-9| cannot be greater than |x|+9, but it can be equal, if x and and 9 are of the same sign. The 9 is negative, so the x must also be negative, giving the solution [tex]
x \le 0
[/tex]


Why is it that the answer book says the answer is [tex]
x < 0[/tex]? Why is it excluding zero? Or is it just wrong?

Thanks,
Dan.
 
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  • #2
All I can suggest is that you go back and check the problem again. If there really is a "[itex]\ge[/itex]" sign rather than just >, obviously x= 0 is a solution. Either you miscopied the problem or the book's answer is wrong.
 
  • #3
I have definately copied the problem correctly from the book.
 
  • #4
This would have many solutions... first x < 0, 0< x < 9 and with x > 9.
 

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