Today in class my teacher said that when solving an inequality where the unknown is in the denominator such as: $$\frac {1}{x} < 3$$
You need to follow these steps:
$$x\neq 0 $$
Solve the equation $\frac {1}{x} = 3$, which is $x=\frac{1}{3}$
Then test the the different "regions" on the number line (i.e between 0 and 0.3, greater than 0.3 or less than 0 and see if the inequality holds true).e.g.
Test x = -1, $-1<3$ (Yes)
Test x = 0.25, $4<3$ (No)
Test x = 1, $1<3$ (Yes)
$$\therefore x<0, x>\frac{1}{3}$$
Is there another way to solve this type of inequality without having to test the various values?
Thanks
Best Answer
You can split it into cases, but it's trickier.
Suppose $x>0$. Then $\displaystyle{1\over x}<3$ implies that $1<3x$, or $\displaystyle x>{1\over3}$.
Suppose $x<0$. Then $\displaystyle{1\over x}<3$ implies that $1>3x$ or that $\displaystyle x<{1\over3}$.
Thus, $\displaystyle{1\over x}<3$ when $x>0$ and $\displaystyle x>{1\over3}$, or when $x<0$ and $\displaystyle x<{1\over3}$. This simplifies to $\displaystyle x>{1\over3}$ or $x<0$.