Challenge Level

Suppose that we are told that four numbers $a, b, c, d$ lie between $-5$ and $5$. Suppose also that the numbers are constrained so that

$$5< a+b < 10 \quad\mbox{ and }\quad -10< c+d < -5$$

Given this information, what can you deduce about these inequalities?

$$ ?? < a+ b- c - d < ?? $$ $$ ?? < a- c < ?? $$ $$ ?? < a - c + d - b < ?? $$ $$ ?? < abcd < ?? $$ $$ ?? < \frac{|a|+|c|}{2}-\sqrt{|ac|} < ??$$

Did you know ... ?

There are many useful general inequalities in mathematics, such as the AM-GM, Cauchy-Schwarz and Jensen's inequalities. These general inequalities are powerful tools which greatly simplify a wide variety of problems in mathematics, in applications from integration to probability via linear algebra.

There are many useful general inequalities in mathematics, such as the AM-GM, Cauchy-Schwarz and Jensen's inequalities. These general inequalities are powerful tools which greatly simplify a wide variety of problems in mathematics, in applications from integration to probability via linear algebra.