Ballparking: Overview

If $$\frac{1}{x-3} > \frac{5}{8}$$, which of the following can be the value of $$x$$?

Good work! [[snippet]] Since $$\frac{5}{8}$$ is positive, $$\frac{1}{x-3}$$ is greater than a positive number, which means $$\frac{1}{x-3}$$ itself has to be positive. Thus, $$x-3$$ itself must be positive, which means that $$x > 3$$. Now that's a wide ballpark, but you can still __POE__ answers that are out of the ballpark. Answer choice A, $$\frac{16}{5} = 3.2$$, is the only answer choice that is greater than 3. Of course you need to actively __POE__ all other answer choices, but that's easily done: they are ordered in descending order. And the next answer choice, $$\frac{12}{5}= 2.4$$ is already smaller than 3.

Incorrect. [[Snippet]] When you look at the inequality $$ \frac{1}{x-3} > \frac{5}{8}$$ which of the following is true?

Incorrect.

Correct

Since $$\frac{5}{8}$$ is positive, $$\frac{1}{x-3}$$ is greater than a positive number, which means $$\frac{1}{x-3}$$ itself has to be positive. Thus, $$x-3$$ itself must be positive, which means that $$x > 3$$. So you looked at the expression and determined that $$x$$ has to be more than 3. Now you don't know if it's 3.5 or 58 or any other number, but you do know for sure that any answer that claims $$x$$ is smaller than 3 should be eliminated. Now go back and look at all the answer choices.

$$\frac{16}{5}$$

$$\frac{12}{5}$$

$$2$$

$$\frac{8}{9}$$

$$\frac{4}{5}$$

$$ \frac{1}{x-3}$$ must be positive.

$$ \frac{1}{x-3}$$ must be negative.

$$ \frac{1}{x-3}$$ can be positive or negative.