Solving two absolute value variables.

[fatcrispy]

Homework Statement

Given: |A|+A+B=15 and A+|B|-B=13. What is A+B equal to? Give all possibilities.

Homework Equations

The Attempt at a Solution

I solve for both absolute variables. So,

A=15-A-B or A=A+B-15
and
B=13-A+B or B=A-B-13

Firstly, I solve for A.

A+A=15-B
2A=15-B
A=\frac{15-B}{2}

and
A-A=B-15
0=B-15
B=15

Plug them back into the original eq. to see if it works.
|\frac{15-B}{2}|+\frac{15-B}{2}+B=15
|\frac{15-B}{2}|=15-B-\frac{15-B}{2}
|\frac{15-B}{2}|=+/- \frac{15-B}{2}
So that works. But when I plug in B=15 into the original eq. it doesn't.

So far, I have A=\frac{15-B}{2}. Next, I solve for B.

B-B=13-A
0=13-A
A=13

and
B+B=A-13
2B=A-13
B=\frac{A-13}{2}

So, I plug in B into the original equation(the second one given).
Plugging in A=13 just comes out to B=+/- B.
Plugging in B=\frac{A-13}{2} results in
\frac{A-13}{2}=+/- \frac{A-13}{2}.

So, when the problem asks for A+B, how do I add them? Do I do:

A+B=\frac{15-B}{2}+\frac{A-13}{2} ?

Please help! Thanks.

 

[Mark44]

I would approach this problem by breaking it up into four cases, to get rid of the absolute values.

I: Assume A > 0 and B > 0.
In this case, the equations are 2A + B = 15 and A + B - B = 13. The 2nd equation is equivalent to A = 13. Solving for B, I get B = -11, which is a contradiction with the assumption that B > 0.

II: Assume A > 0 and B < 0.
With this assumption |A| = A and |B| = -B.
This case gives me values for A and B that don't contradict the assumption in this case.

III: Assume A < 0 and B > 0.
IV: Assume A < 0 and B < 0.

 

[fatcrispy]

I would approach this problem by breaking it up into four cases, to get rid of the absolute values.

I: Assume A > 0 and B > 0.
In this case, the equations are 2A + B = 15 and A + B - B = 13. The 2nd equation is equivalent to A = 13. Solving for B, I get B = -11, which is a contradiction with the assumption that B > 0.

II: Assume A > 0 and B < 0.
With this assumption |A| = A and |B| = -B.
This case gives me values for A and B that don't contradict the assumption in this case.

III: Assume A < 0 and B > 0.
B=15, A=13. Plugging in shows that A does equal 13 but B=-11 which contradicts B>0.
IV: Assume A < 0 and B < 0.
I get B=15, B=(A-13)/2. Plugging in I get A=13(contradicts A<0) and B=-11.

So, it seems that II is the only one without a contradiction. So, does that mean I only use A=(15-B)/2 and B=(A-13)/2 ?? And just add them A+B=(15-B)+(A-13) / 2. So, my final answer will be (A-B+2)/2 = A+B?

 

[fatcrispy]

Right? (A-B+2)/2 gives me the value of A+B? Nothing else would be the value right? I appreciate your help.

 

[Mark44]

No, your final answer should be a number. What values did you get for A and B in the 2nd case?

 

[fatcrispy]

No, your final answer should be a number. What values did you get for A and B in the 2nd case?

Oh, well in that case I got A=43/5 and B=-11/5. Thus, A+B=32/5 ? and that is the only answer right?

 

[fatcrispy]

Could someone please help me finish this problem? Mark, thanks for everything!

 

[Mark44]

Those are the numbers I get for the 2nd case. The only cases I checked are the first two, so if you're confident in your results for the 3rd and 4th cases, then you've got your answer.