Solving equation containing absolute values

[jkristia]

Homework Statement

I have this equation

|2x+7| - |6-3x| = 8.

The step I did is to replace the || with () and then solve the equation
2x+7-6+3x = 8
X = 7/5

But how do a go about solving for the second solution?
With one absolute value I would
|2x + 7| = 8
2x + 7 = +-8
2x = -7 +-8
X = -15/2, x = ½

But I can’t see how to solve the first equation for the second solution.

 

[Mentallic]

The best way to approach this type of problem is to take 3 cases.

Firstly, you need to find at which x value each absolute value is positive or negative. 2x+7>0, then x>-7/2, thus for x>-7/2 that expression is positive and for x<-7/2 that expression is negative.
6-3x>0, x<2, thus for x<2 it is positive, and x>2 it is negative.

Now the first case you should consider is for all x values such that both expressions are positive.
The next case should be for x values where one is positive and the other is negative.
The last case should be when both are negative.

As an example, if we have to solve |x|+|x-1|=2, for x>1 both are positive, so we simply solve x+(x-1)=2, for 0<x<1 we have the first being positive and the second being negative, thus we solve x-(x-1)=2 and for x<0 both are negative so we solve -x-(x-1)=2.

Just apply the same idea to your question.

p.s. remember that since we assumed x<0 for the last case, the solution needs to be less than zero, else it is an invalid solution and you just discard it.

 

[jkristia]

I think I got it, thank you for your help.