absolute

[maihuna]

||x-9| -3| = 6 has how many solutions:

0
1
2
3
4

[truplayer256]

|x-9|-3=6
-|x-9|+3=6
|x-9|=9--> -x+9=9, x-9=9 x=0,x=18
-|x-9|=3--> |x-9|=-3--> -x+9=-3, x-9=-3 x=12, x=6 (These two solutions don't work when inputted into the original problem)

A total of 2 solutions.

[maihuna]

||x-9| -3| = 6
=> |x-9|-3 =6 => for x<9 -(x-9)-3 = 6-x = 6 =>x=0
for x>9 x-9-3 = x-12=6 => x=18

and |x-9|-3 = -6 =>for x<9 6-x=-6 or x=12 not a soln as x<9
for x>9 x-12 = -6 or x=6 not a soln as x>9

So two solns

[deepoe]

Isn't it 4 solutions?? Because you can divide the -3 or +3 ?

[rossmj]

I think the best way to look at this one is using a number line. Inside the the absolute values you can see the lx-9l is the distance x is from nine, then we are trying to find the distance of this number from three, which we are told is 6. There are only 2 numbers that are exaclty 6 spaces away from 3 on the number line. 9 and -3 so we now know that lx-9l must result in one of these two numbers which gives us two values for x 0 or 6.

[vittalgmat]

Two solutions.
I followed the numberline method coz it is much more intuitive and faster (atleast for this problem)

[Ian Stewart]

||x-9| -3| = 6 has how many solutions:

0
1
2
3
4

I'd also use a 'distance' approach here, but there is one important observation we need to make - |x - 9| can't be negative. If we let y = |x - 9| and look at an equation like the following:

|y - 3| = 6

then the distance between y and 3 is equal to 6, so y can only be -3 or 9. So y = |x - 9| = -3, or |x - 9| = 9. Since it's impossible for |x - 9| to be negative, the only possibility is that |x - 9| = 9, or that "the distance between x and 9 is 9". So x could be 0 or 18, and we have two solutions.

[4seasoncentre]

Isn't it 4 solutions?? Because you can divide the -3 or +3 ?

I thought so too, but then I tried playing with it.

So
| |x-9| -3| = 6
means that...
A) |x-9|-3 = 6 and
B) |x-9|-3 = -6

From A:
|x-9| = 9
(x-9) = 9 or (x-9) = -9
x = 0 or x = 18

From B
|x-9| = -3
and I say to myself... whoa... absolute values can never equal a negative. Stop here.

I found 2 solutions.

[maihuna]

||x-9| -3| = 6 has how many solutions:

0
1
2
3
4

I'd also use a 'distance' approach here, but there is one important observation we need to make - |x - 9| can't be negative. If we let y = |x - 9| and look at an equation like the following:

|y - 3| = 6

then the distance between y and 3 is equal to 6, so y can only be -3 or 9. So y = |x - 9| = -3, or |x - 9| = 9. Since it's impossible for |x - 9| to be negative, the only possibility is that |x - 9| = 9, or that "the distance between x and 9 is 9". So x could be 0 or 18, and we have two solutions.

Thats truly amazing Ian. You rocks. Thanks.
Maihuna