Possible Values of a and b for |2x+a| = |b-x| at x=-4 and x=2/3

[danago]

The equation |2x+a|=|b-x| has exactly 2 solutions, at x=-4 and x=2/3. Find the value(s) of a and b.

Ok so the questions is asking me to find possible values of a and b which make the equation true for ONLY x=-4 and x=2/3.

So for:

|a-8|=|b+4|
|a+4/3|=b-2/3|

I need to find the values of a and b that satisfy both equations.

Its from there I am a little stuck. Any help would be greatly appreciated.

Thanks in advance,
Dan.

 

[berkeman]

The effect of the absolute value is to give you two possible equations for each single one that you are given, right? So if you have an equation |X| = |Y|, that expands into two possible equations:

X = Y and -X = Y

So this may be the long way to solve this, but I'd expand out your two equations into four, and combine them in different pairs to see what the potential solutions could be for a and b.

EQN1 --> E1p and E1n
EQN2 --> E2p and E2n

There are four combinations of equations that I think would be valid to use for the solutions. Obviously you can't combine E1p and E1n or E2p and E2n, but the other combinations should be valid to see what solutions come up.

 

[danago]

Ok so my 4 new equations, in which I've isolated 'a',
a=b+12
a=4-b

a=b-2
a=-2/3-b

Of the 4 possible combinations of equations, only 2 have solutions, -19/3 and 1. Using these values to get values of a, i get the two possible solutions of a and b:

a=17/3
b=-19/3

a=1
b=3

Seems to fit the requirements :) Thanks for the help.