Absolutely Less Than Zero - wrong explanation

[ChrisPF]

Q from CAT: Absolutely Less Than Zero

Part of the explanation seems wrong:

|x - 3| = |2x - 3|

Now let's consider what happens when only one side is negative; in this case, we choose the right-hand side:
|x - 3| = |2x - 3|
x - 3 = -(2x - 3)
x - 3 = -2x + 3
3x = 6
x = 2

We can verify that this is a valid solution by plugging 2 into the original equation:
|2 - 3| = |2(2) - 3|
|-1| = |1|
1 = 1
---------------

Although x=2 is a soln it is not a soln when |2x - 3| is -ve as x<3/2
So explanation wrong?

[RonPurewal]

Although x=2 is a soln it is not a soln when |2x - 3| is -ve as x<3/2
So explanation wrong?

i had to read that a few times, but yes, i see what you're saying, and you have a valid point.

you're saying that the solution mistakenly portrays (x - 3) as a positive quantity and (2x - 3) as a negative quantity, when the opposite is in fact true. and you are 100% right.

--

let me explain why the method nevertheless produces the correct solution.

FACT: there are only two possible outcomes in an equation setting two absolute values equal. either (1) the expressions within the absolute-value bars are identical, or (2) the expressions have the same magnitude but opposite signs. (if you don't see why this is true, then ponder a simpler equation, such as |x| = |y|.)

because of this FACT, you only need to solve 2 different equations to guarantee a complete solution:
(a) the equation produced by simply stripping the absolute-value bars off the expressions, and not changing anything else (which corresponds to possibility (1) above), and
(b) the equation produced by stripping off the absolute value signs and then reversing the signs on one side of the equation (which corresponds to possibility (2) above).

when you solve (b), it's completely immaterial which side you choose for reversing the sign, as x = -y and -x = y are the same equation. in this particular example, the choice of the right-hand side turned out to be 'retroactively wrong', so to speak, but that doesn't matter at all. in fact, it would be pointless to worry about reversing the 'correct' side of the equation, as doing so would double the amount of work while still finding the same solutions.

hth!

[ChrisPF]

Well the explanation leads with "Here, again, we must consider the various combinations of positive and negative for both sides"

Anyway, useful tip. Thanks
Chris

[StaceyKoprince]

Yes, we're re-writing the explanation to make this more clear (in fact, I think Chris may already have done so). Thanks for the heads-up!