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What interval consists of all u such that neither 2u nor -20u is in the interval \((-10,10)\)?

 May 15, 2022
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If 2u is not in the interval, then \(2u \leq -10 \text{ or }2u \geq 10\). That means \(u \leq -5 \text{ or }u \geq 5\).

 

Moreover, -20u is not in the interval, so \(-20u \leq -10 \text{ or }-20u \geq 10\). That means \(u \leq -\dfrac12 \text{ or }u \geq \dfrac12\).

 

If \(u \leq -5 \text{ or }u \geq 5\), then it is true that \(u \leq -\dfrac12 \text{ or }u \geq \dfrac12\). So \(u \leq -\dfrac12 \text{ or }u \geq \dfrac12\) is extraneous. The range of values of u is \(u \leq -5 \text{ or }u \geq 5\). Expressed in interval notation, \(u \in (-\infty, -5]\cup [5, \infty)\).

 May 15, 2022

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