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3( x + 6) = 9x - 6

3( x + 6) = 9x - 6

The parentheses are supposed to be
absolute value signs. I know that there are two answers to this problem. I know that one of the answers is positive 4. I thought that the other answer to this was -2; however, my teacher said it was wrong. HELP!!!!!!!!!
2 REPLIES 2

3( x 6) = 9x - 6

3|x+6|=9x-6

To solve such a problem, you must exploit the specific properties of the absolute value.

What does the absolute value do to positive numbers?

|5| = 5
|12| = 12 ----- The absolute value does nothing to positive numbers. Just for the record, absolute value does nothing to zero (0), either.

What does the absolute value do to positive numbers?

|-4| = 4
|-11| = 11 ----- The absolute value changes the sign on negative numbers.


Now think of |x+6|

x+6 is positive or zero OR negative. What we need it to know WHEN it falls in these two classes.

x+6>=0, then (subtract 6)
x>=-6, and
|x+6| = x+6 --- The absolute value did nothing to a number we know to be positive or zero.

Then solve this case:

3|x+6|=9x-6
3(x+6)=9x-6
3x+18=9x-6
3x+18+6=9x
3x+24=9x
24=9x-3x
24=6x
(24/6)=x
x=4


Okay, now for the other:
x+6<0, then (subtract 6)
x<-6, and
|x+6| = -(x+6) --- The absolute value changed the sign of a number we know to be negative.

Then solve this case:

3|x+6|=9x-6
3(-(x+6))=9x-6
3(-x-6)=9x-6
-3x-18=9x-6
-3x=9x-6+18
-3x=9x+12
-3x-9x=12
-12x=12
x=(12/(-12))
x=-1

So, are we done? Not quite. We have to results, but do they make sense.

First, we promised

x>=-6

This led to the solution

x=4

Is 5>=-6? Yes it is. This is likely to be a solution.

Second, we promised

x<-6

This led to the solution

x=-1

Is -1<-6? No it is not. This cannot be a solution, since the problem doesn't even exist at x=-1. We promised x<-6 for this piece!

Anyway, the cardinal rule of ANY problem is to check your answers in the ORIGINAL problem statement.

3|x+6|=9x-6

Try x=4

3|4+6|=9(4)-6
3|10|=36-6
3(10)=30
30=30 ---- Okay. That works.

Try x=-1
3|(-1)+6|=9(-1)-6
3|5|=-9-6
3(5)=-15
15=-15 ------- WHAT???!!!
I guess x=-1 doesn't work.

Proceed with faith. You'll get it.

3( x 6) = 9x - 6

Did you test x=-2? That will end any dispute.

3|x+6|=9x-6
3|(-2)+6|=9(-2)-6
3|4|=-18-6
3(4)=-24
12=-24 ------------ Perhaps not!

A result this silly can come from only two sources:

1) I did something wrong in my algebra, or
2) We started with something that had no chance.

I'm pretty confident on my algebra. Must be the value x=-2 that is the problem.

End of argument.
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