Hi, first thanks for your help, and for the page!, and sorry for my English, I'm from Argentina

I was looking the example about the absolutes, and I find |x–3|=|3x+2|–1 that's similar what I'm searching (and when I put resolve graphically the example in the page it plotted me another thing, I send a message asking for that).

I put the expression |4x|=|4x+1| in plotters online but tell me that's an invalid expression, and in the plotter of this page it show me another graphic too.

Hope you can help me, telling me if both resolutions are done (Isn't the first way more easy?), and the arguments are fine. At the end I ask about the absurds finded in the solution, I remain dobious about it.

When I solve the exercice I put

|4x|=|4x+1|

4x = + - |4x+1|

that give me 4x = 4x+1 , -4x=4x+1 (both in case +|4x+1| when I take the bars of)

And +/- (-4x) = 4x + 1 when the case is -|4x+1| by multiplying for -1 both sides, and taking the bars off. That give me 4x=4x+1 and -4x=4x+1 the same cases above

In the case of 4x=4x+1 -> 0=1 that's an absurd or ilegal expression don't?

In the case of -4x=4x+1 -> -8x=1 -> x=-1/8

That's the same solution if I do by the method explained here, looking the intervals

|4x| >= 0 for x>= 0 doing 4x >= |0| and then x >=0/4 then x>=0

|4x+1| >= 0 for x>= (-1/4)

I know Y=4X has a positive slope and Y=4X+1 too.

That's points, 0 and -1/4, are where the absolute-value expressions equal zero, giving me the intervals, (-infinity,-1/4),(-1/4,0),(0,infinity).

In the first interval, both absolute values are negatives, so I have -4x=-4x-1 -> 0=-1 that's illegal

the same thing happens in the last interval 0=1

but in the middle -4x=4x+1 then -8x=1 and then x=(-1/8) that's between -1/4 and zero and is a valid solution.

Then the answer is X=(-1/8), the same above.

That's fine? what's means I have an absurd? in the example in the page get an invalid solution because they have out of interval solution, but I get an absurd, that's fine? the solutions are the points of intersections of the lines, but the invalid solutions are nothing in the graphic it seems?

Thanks, and sorry for the extension!