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'Weekly Challenge 1: Inner Equality' printed from http://nrich.maths.org/

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Since $-5 < a, b, c, d < 5$ the inequalities $5< a+b< 10$ and $-10< c+d< -5$ show that
$$
0< a, b< 5\quad\quad -5< c, d < 0
$$
 
It is possible then to conclude that
 
 $$ 10 < a+ b- c - d < 20 $$
$$ 0 < a- c < 10 $$
$$ -10 < a - c + d - b < 10 $$
$$ 0 < abcd < 625 $$
$$ 0 < \frac{|a|+|c|}{2}-\sqrt{|ac|} < 2.5$$
 
Note that the lower bound of the fourth inequality could be deduced from the AM-GM inequality for two numbers.
 
Note also that, since it is not possible to set, for example, $a=5$ care must be taken to construct a really clear justification of the results.