How do you solve this absolute value eq: 8|x+8| + 9 > 6?

Question

I'm having problems with this absolute inequality: 8|x+8| + 9 > 6 =

Solve for x (obviously)

I get mostly confused after I get rid of the 9, but what am I supposed to do with the 8 outside the absolute value signs?

Thanks! 10 pts. for clearest directions on how to get the answer. :)

Answer 1

As you said you first get rid of the 9 by subtracting 9 on both sides of the inequality.  Subtracting does not change the direction of an inequality.

8 |x + 8 | + 9 > 6

<==>  8 |x+8| > -3

Then you can divide by 8.  Division/multiplication by positive numbers do not change the direction of the inequality (but if you multiply/divide by a negative number, then the you have to flip the direction).  Think about if you had more pieces of candy than your friend and you both eat the same fraction of it (half for example), then you still have more pieces of candy than your friend.

8 |x+8| > -3

<==>  |x+8| > -3/8

Now, here you know that the absolute makes every thing positive.  So no matter what x is |x+8| >= 0 > -3/8

So this inequality holds for any real number x.

If you think about it with regards to the original equation, no matter was x you choose, |x+8| >= 0 so 8*|x+8| >= 0 so 8*|x+8| + 9 >= 9 > 6

It always holds.