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# Inequaity

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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