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Why if I squared this equation $$|4x-3|=-|2x-1|$$ it will give a solution of $\frac{2}{3}$ and $1$?

But actually, it has no solution. Why squaring the equation will cause an error?

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  1. Because from your equation we have: $$4x-3=2x-1=0,$$ which is impossible.

  2. Because $a=b\Leftrightarrow a^2=b^2$ is wrong.

The counterexample is your equation. Also, for example. $-1=1$ is wrong, but $(-1)^2=1^2$ is true.

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Try to substitute thsoe values inside to see what is going on

if you substitute $x=1$. you get $LHS=1$, $RHS=-1$, clearly $1 \neq -1$ but $1^2=(-1)^2$.

You can still use squaring to obtain possible solutions, however, you have to check whether they still satisfy the original problem.

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Squaring both sides and solving for $x$ is like solving $f(|4x-3|)=f(-|2x-1|)$ where $f(x)=x^2.$ This may yield an extraneous "solution" since $f(x)=x^2$ isn't one-to-one i.e. $$f(|4x-3|)=f(-|2x-1|) \nrightarrow |4x-3|=-|2x-1|.$$ If you seek to obtain an equation with the same solution set as the original, make sure you're working with one-to-one functions.

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