Multiple choice

If $x \, \varepsilon \ $ $\displaystyle [0,2\pi]$, then t he equation $|sin x | = sin x +3$ has

  1. no root

  2. only one root

  3. two roots

  4. more than two toots
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A Correct answer
Explanation

For every real x, |sin x| is at least sin x. Therefore, |sin x| = sin x + 3 would require |sin x| - sin x = 3. The left side can only range from 0 to 2, so there are no roots.

AI explanation

We can rewrite the absolute value equation |sin x| = sin x + 3 by considering both possible cases for the sign of sin x. If sin x is positive or zero, the equation simplifies to sin x = sin x + 3, which results in 0 = 3 and is a contradiction. If sin x is negative, the equation becomes -sin x = sin x + 3, yielding sin x = -1.5, but this is impossible since the range of the sine function is limited to [-1, 1]. Since no scenario produces a valid solution, the equation has no root.

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