Darius, Richard, and Libor solved an equation with absolute value: $$ |x| = x -1. $$ Each of them solved the equation in their own way:
** Darius ** remembered that from the equation $|x|=9$, it follows $x=9$ or $x=−9$, so he decided to proceed analogously: $$ x=x−1 \mathrm{~or~} x=−(x−1). $$ He solved both equations in his head and found that the first equation has no solution ($0\neq−1$) and the second equation has the solution $x=\frac12$. Darius then made a check, which showed that $x=\frac12$ does not satisfy the original equation. He concluded that the given equation has no solution.
Richard remembered that $|x|^2=x^2$, so he decided to eliminate the absolute value by squaring both sides of the equation: $$ (|x|)^2=(x−1)^2. $$ He then proceeded as follows: $$ \begin{align} x^2=x^2−2x+1 \cr 2x=1 \cr x=\frac12 . \end{align} $$ Richard also came to the solution $x=\frac12$. He made a check and found that the given equation has no solution.
Libor followed the definition of absolute value, i.e., for any real number: $$ |x|=x \mathrm{~if~} x\geq 0 \mathrm{~and~} |x|=−x \mathrm{~if~} x<0. $$ In the first case, for $x\geq 0$, he got the equation: $$ x=x-1\mathrm{~(no~solution)} $$ In the second case, for $x<0$, he got the equation $-x=x-1$, i.e., $$ \begin{align} x=-(x-1) \cr x=\frac12. \cr \end{align} $$ After making the check, he stated that the given equation has no solution.
Is any of the solutions presented by the students wrong?
No, all solutions are correct.
Yes, Richard's solution is wrong. It should have been $|x|^2=\pm x^2$
Yes, Darius's solution is wrong. We can use this reasoning only if there is a number on the right side of the equation.
Yes, Libor's solution is wrong. He should have solved the equation for $x \in (-\infty;1)$ and for $x \in [ 1;+\infty)$.