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For, the problem $|x + 1| = |2x - 1|$, I found one solution analytically:

$x + 1 = 2x - 1$

$\to x = 2$

Since, these are absolute functions, they should intersect once more at $x = 0$. This solution I got by guessing integers. Is it possible to find this solution analytically? My textbook says to graph the functions and find the other solution.

Dstarred
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    One could raise both sides to the second power (which would allow to remove absolute sign) and then factor. – Salcio Dec 06 '21 at 00:39
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    If you consider what equations you could take the absolute value of to get $|a|=|b|$, you find that either $a=b$ or $a=-b$. (Or the negations of these). You've considered the $a=b$ case, and the $a=-b$ case solves rather easily to $x=0$ – Angelica Dec 06 '21 at 00:47
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    @Angelica Thanks ... so the rule to find solutions for absolute functions: "$a = b$ or $a =−b$" can be applied to entire equations too? – Dstarred Dec 06 '21 at 01:04
  • @Salcio thanks for your help too – Dstarred Dec 06 '21 at 01:05

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If two expressions have the same absolute value, then either they are equal or one is the negative of the other.

So either $x+1=2x-1$ or $x+1=-2x+1$.

So the two solutions are $x=2, x=0$.

John Wayland Bales
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