How Does This Work?

Algebra Level 2

$a > 1 > b > 0 > c$

Given the above chain of inequalities, which of the following statements must be true?

bc > 0

ac > b^2

bc > ac

ab > 1

1 solution

Jesse Nieminen
Jul 23, 2016

$ab > 1$ is not necessarily true. e.g. $2 \times \dfrac{1}{2} > 1$ is false.
$bc > 0$ is always false since $b > 0$ and $c < 0$ .
$ac > b^2$ is always false since $ac < 0$ and $b^2 > 0$ .
$bc > ac$ is always true since $b < a$ and $c < 0$ and multiplying both sides of an inequality by negative flips the order.

Hence, the answer is $\boxed{bc > ac}$ .

another way to prove the true case is: we are given that $b<a$ . Multiply both sides by $c$ , which is negative, so reverse the inequality sign, thus $bc>ac$ .

Richard Costen - 4 years, 10 months ago

This is the way I proved it to be true.

Jesse Nieminen - 4 years, 10 months ago

Great observations. We cannot "multiply inequalities" together and hope that the sign stays the same. We do need to check what could happen under different conditions.

Calvin Lin Staff - 4 years, 10 months ago

How Does This Work?

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