The complex numbers $z$ which satisfy the inequality ${log}_{1 / 3} | z + 1 | > {log}_{1 / 3} | z - 1 |$ are such that their real part is $-$ ive. Is this statement true?

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Solution

Yes, true. Since base $1 / 3$ is less than $1$ , so given
inequality holds if
$| z + 1 | < | z - 1 |$
or ${(x + 1)}^{2} + y^{2} < {(x - 1)}^{2} + y^{2}$
or $4 x < 0$ or $x < 0$ or $R e (z) < 0$

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