Question

So I'm having a very difficult time figuring this out:

4|x-3|+2|x+3|=12

The | is absolute value. I know that if this was like |x-3|=|5| or something like that, it would be 4 answers, but I have no idea how to solve the one above.

Answer 1

You can solve it the ~same way, subtract one of the abs values from both sides, like so:
4|x-3| =12 - 2|x+3|
then divide by 4, drop the abs value signs on the left, making two equations (one where the right is positive, another where the right is negative). Then isolate the absolute value which is still on the right, and repeat.

You may need to do it again, starting from
2|x+3| =12 - 4|x-3|

Answer 2

I hate to waste your time, but could you demonstrate that? I'm not fully grasping that concept.

Answer 3

After dividing the first by 4:
|x-3| =3 - 0.5|x+3|
then drop the signs:

x-3 =3 - 0.5|x+3| and x-3| = -(3 - 0.5|x+3|)

From here, isolate the absolute value again, and solve.

Answer 4

There is also another way. I put it, in case it helps you. You can get rid of absolute values, in the following way,
If x<(-3)
\[\text{If}\ \ x<-3\ \ \ \ -4(x-3)-2(x+3)=12\]
\[\text{If}\ \ 3>x>-3\ \ \ \ -4(x-3)+2(x+3)=12\]
\[\text{If}\ \ x>3\ \ \ \ 4(x-3)+2(x+3)=12\]
Solve this three equations, but obtain the solutions in the regions indicated.

Answer 5

And you can also graph it, by subtracting 12 on both sides and finding where it equals zero (you could check your solution this way)

4|x-3|+2|x+3| - 12 = 0
https://www.google.com/search?safe=off&sclient=psy-ab&q=4%7Cx-3%7C%2B2%7Cx%2B3%7C&btnG=#q=4ABS(x-3)%2B2ABS(x%2B3)-12&safe=off

Answer 6

Thank you, I'll check this out and see if I need any other help.

Answer 7

John, is this method used to solve regular absolute values like |x-3|=|5|?

Also... this is an inequality... how does that relate to the equation if you're setting x<-3, 3>x>-3, x>3, etc.?

Answer 8

Yes it can be used for these inequalities. You only need to check where the function inside the absolute value changes its sign.

Answer 9

In the case you cite, you know x-3=0, implies that x=3. So there is a change of sign in 3. Then,
\[x<-3\Rightarrow -(x-3)=5\]
\[x>-3\Rightarrow (x-3)=5\]

Answer 10

x<-3, 3>x>-3, x>3, etc.?

it just means you need to get a value of x from that particular equation that fits those domains. If you get an x that doesn't fit the domain for the particular equation, discard it.

eg

If x<−3 −4(x−3)−2(x+3)=12
this gives x=-1... this doesn't fit the domain x<-3, so is not a valid solution to the original abs value equation.

Answer 11

Thanks