1. ## inequality in intervals

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2. $\displaystyle \frac{1}{x} < 4$.

First, it should be abundantly obvious that $\displaystyle x \neq 0$.

To solve this inequality for $\displaystyle x$, you will need to consider two cases, the first being when $\displaystyle x$ is positive, and the second when $\displaystyle x$ is negative, since multiplying/dividing by a negative reverses the inequality sign.

Case 1: $\displaystyle x > 0$.

$\displaystyle \frac{1}{x} < 4$

$\displaystyle 1 < 4x$

$\displaystyle \frac{1}{4} < x$.

Since $\displaystyle x > 0$ and $\displaystyle x > \frac{1}{4}$, putting it together gives $\displaystyle x > \frac{1}{4}$, or $\displaystyle x \in \left(\frac{1}{4}, \infty\right)$.

Case 2: $\displaystyle x < 0$

$\displaystyle \frac{1}{x} < 4$

$\displaystyle 1 > 4x$

$\displaystyle \frac{1}{4} > x$.

So $\displaystyle x < 0$ and $\displaystyle x < \frac{1}{4}$. Putting it together gives $\displaystyle x < 0$, or $\displaystyle x \in (-\infty, 0)$.

Therefore, our final solution is

$\displaystyle x \in (-\infty, 0) \cup \left(\frac{1}{4}, \infty\right)$. You can check this by graphing the functions $\displaystyle y = \frac{1}{x}$ and $\displaystyle y = 4$ and making sure that everything below the line $\displaystyle y = 4$ are the $\displaystyle x$ values listed above.