Understanding Solutions to Inequalities Involving Absolute Values

[danago]

Solve the inequality $$\left| {x - 9} \right| - \left| x \right| \ge 9$$

I started by rewriting it as:
$$\left| {x - 9} \right| \ge 9 + \left| x \right|$$

Now, for any real numbers x and y,
$$\left| {x + y} \right| \le \left| x \right| + \left| y \right|$$

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 $$x \le 0$$


Why is it that the answer book says the answer is $$x < 0$$? Why is it excluding zero? Or is it just wrong?

Thanks,
Dan.

 

[HallsofIvy]

All I can suggest is that you go back and check the problem again. If there really is a "$\ge$" sign rather than just >, obviously x= 0 is a solution. Either you miscopied the problem or the book's answer is wrong.

 

[danago]

I have definitely copied the problem correctly from the book.

 

[Werg22]

This would have many solutions... first x < 0, 0< x < 9 and with x > 9.