How to solve this inequality?

0 < |x + 3| < 6

I would split this into two inequalities:

0 < |x + 3| and |x + 3| < 6

Just looking at the first one, for all values of x other than -3, this is true, since absolute value is never negative, so that's an always true condition, given that x ≠ -3.

so now we'll look at the other one. If this was an equation, I'd remove the absolute value and put a ± on the other side. Since this is an inequality, you still perform the same action, but the - side has to have the sign flipped. So we can split this into two inequalities to have:

x + 3 < 6 and x + 3 > -6

Now solve each:

x < 3 and x > -9

So we have the range of:

-9 < x < 3

Taking into account that x ≠ -3, our solution are the following two ranges:

-9 < x < -3 and -3 < x < 3

|x + 3| > 0 will be true for any x as long as x + 3 is nonzero. So in other words, this part is true for x not equal to -3.

|x + 3| < 6 is equivalent to -6 < x + 3 < 6.

Subtract 3 from each "side" and you get

-9 < x < 3

So combining those two conditions, this inequality is true for all x in the above interval, except x = -3.

|x + 3| &gt; 0 shall be true for any x as long as x + 3 is nonzero... so in other words, this part is true for x not equal to -3...

|x + 3| &lt; 6 is equivalent to -6 &lt; x + 3 &lt; 6...

subtract 3 from each "side" and you get

-9 &lt; x &lt; 3

so combining those two conditions, this inequality is true for all x in the above interval, except x = -3...

We have that : 0 <| x + 3 | < 6 means that

: | x + 3 | > 0 AND | x + 3 | < 6

Solve one at a time:.............. | x + 3 | > 0

then x + 3 > 0 OR x + 3 < 0 that is x can nob be - 3

| x + 3 | < 6 .. means that : - 6 < x + 3 < 6 ....................... - 9 < x < 3

Since x can not be -3 the solution is :

- 9 < x < 0 OR 0 < x < 6

0 < x + 3 < 6 or -6 < x + 3 < 0

-3 < x < 3 or -9 < x < -3

So x can be anything between -9 and 3 (exclusive) except -3.

0 < |x + 3| < 6

-6 < x + 3 < 6

-9 < x < 3

0 < x + 3 < 6

-3 < x < 3

-6 < x + 3 < 0

-9 < x < -3

(-9, -3) U (-3, 3)