State True and False
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 -ve .

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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 $R e (z) < 0$

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{log}_{1 / 3} | z + 1 | > {log}_{1 / 3} | z - 1 |

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l o g_{1 / 3} | z + 1 | > l o g_{1 / 3} | z - 1 |

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The locus of z satisfying the inequality $l o g_{\frac{1}{3}}$ |z+1| > $l o g_{\frac{1}{3}}$ |z-1| is

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