Inner equality
Can you solve this inequalities challenge?
Age
16 to 18
Challenge level
Problem
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.
Getting Started
Inequalities are an important extension of algebra which are needed more formally in C1 and beyond.
Note that some, but not all, algebraic manipulations still work with equals signs 'replaced' by inequality signs -- you need to take extra care when algebraically manipulating inequalities.
Addition or subtraction of a quantity is straightforward with inequalities. For example, if we know that $5< a+b< 10$ then we know that $5-b< a< 10-b$. However, we need to take more care with division and multiplication as minus signs cause inequalities to reverse under these operations.
Direct algebra will not help you much in this problem. You will have to make deductions such as 'if a is a very small positive number than b must be very close to 5'.
Writing down such statements is difficult to do clearly, so focus on the inequalities intuitively if need be.
Student Solutions
Well done to Amrit, Adithya, Daven and Sergio who all sent in solutions to this problem. Here are the first four inequalities:
$$ 10 < a+ b- c - d < 20 $$
$$ 0 < a- c < 10 $$
$$ -10 < a - c + d - b < 10 $$
$$ 0 < abcd < 625 $$
Aditha's solution explains how to get each of them, you can read the pdf
Amrit explained how to work out the fifth inequality:
For the last inequality, we need to prove the AM-GM inequality
$\frac{a+c}{2}>\sqrt{ac}$
$a+c>2\sqrt{ac}$
$a^2+2ac+c^2>4ac$
$a^2-2ac+c^2>0$
$a-c)^2>0$
A number squared is always greater than 0 unless the number is 0 or in this case if a=c
Plugging in a=c into the last inequality, we have
$\frac{|a|+|c|}{2}-\sqrt|ac|>0$
Looking at the AM-GM inequality, we want a and c to be as far apart as
possible. So |a| has to be 5 and |c| has to be 0 or vice versa.
Applying this, we have $\frac{|a|+|c|}{2}-\sqrt|ac|<\frac{5}{2}$
$$ 10 < a+ b- c - d < 20 $$
$$ 0 < a- c < 10 $$
$$ -10 < a - c + d - b < 10 $$
$$ 0 < abcd < 625 $$
Aditha's solution explains how to get each of them, you can read the pdf
Amrit explained how to work out the fifth inequality:
For the last inequality, we need to prove the AM-GM inequality
$\frac{a+c}{2}>\sqrt{ac}$
$a+c>2\sqrt{ac}$
$a^2+2ac+c^2>4ac$
$a^2-2ac+c^2>0$
$a-c)^2>0$
A number squared is always greater than 0 unless the number is 0 or in this case if a=c
Plugging in a=c into the last inequality, we have
$\frac{|a|+|c|}{2}-\sqrt|ac|>0$
Looking at the AM-GM inequality, we want a and c to be as far apart as
possible. So |a| has to be 5 and |c| has to be 0 or vice versa.
Applying this, we have $\frac{|a|+|c|}{2}-\sqrt|ac|<\frac{5}{2}$