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If $|z| \geq 3$, then the least value of $\left|z+\frac{1}{4}\right|$ is
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Verified Answer
The correct answer is:
$\frac{11}{4}$
$\left|z+\frac{1}{4}\right|$
$=\left|z-\left(\frac{-1}{4}\right)\right| \geq|z|-\left|\frac{-1}{4}\right|$
$=\left|(-z)-\frac{1}{4}\right| \geq\left|3-\frac{1}{4}\right|=\frac{11}{4}$
$\therefore \quad\left|z+\frac{1}{4}\right| \geq \frac{11}{4}$
$=\left|z-\left(\frac{-1}{4}\right)\right| \geq|z|-\left|\frac{-1}{4}\right|$
$=\left|(-z)-\frac{1}{4}\right| \geq\left|3-\frac{1}{4}\right|=\frac{11}{4}$
$\therefore \quad\left|z+\frac{1}{4}\right| \geq \frac{11}{4}$
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