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When writing the solution to an inequality in interval notation, say $x < 5$ as $(-\infty, 5)$, how can the $x$ be "involved"?

Is it correct to just write $(-\infty, 5)$, or should it be $x=(-\infty, 5)$, or perhaps $x\in(-\infty, 5)$?

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2 Answers

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One defines the interval $(-\infty, a)$ precisely as $\{x:x<a\}$. So saying

$$x\in (-\infty,a)$$ is saying the same as saying

$$x<a$$

Note that writing $x=(-\infty, a)$ would only by correct if you're saying "All elements of $x$ are solution" (viz, $x$ would be denoting an set, not a real number). It is usual to write $S=(-\infty,a)$, where $S$ stand for "solution set", that is, all elements of $S$ are solutions.

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The third option. You are in essence saying that $x$ belongs to the interval between $-\infty$ and $5$, or in other words, $x$ is contained in the set that includs all numbers larger than $-\infty$ and smaller than $5$.

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