Inner equality
Can you solve this inequalities challenge?
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.
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 then 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 $\frac{|a|+|c|}{2}-\sqrt|ac|<\frac{5}{2}$
Teachers' Resources
Why do this problem?
This problem provides students with practice in thinking about inequalities and how they can be combined, considering the effect of addition, subtraction and multiplication. It leads to a simple example of the AM-GM inequality.
Possible approach
Considering just two values which lie between $-5$ and $5$, ask students what they can deduce about the minimum and maximum values of the sum, difference and product of these numbers. Compare their results to the further constraints given on $a, b, c, d$ in the problem. What further can be said about $a$ and $b$, $c$ and $d$? Working in pairs or small groups, students could consider each of the further inequalities, checking their intuition by first trying values within the permitted ranges and then by comparing ideas with another pair or group.
The final inequality (AM-GM) may need additional prompts, time to experiment and algebraic manipulation.
Key questions
What can you deduce about the maximum and minimum possible values of the sum of two numbers between $-5$ and $5$? The maximum and minimum difference? Product?
If the maximum and minimum values of $a+b$ and $c+d$ differ from this, what can you say about the individual values of $a, b, c$ and $d$?
Can the specified quantity ever be zero or negative? Why or why not?
Possible support
Expressions involving just two quantities could be explored graphically using Desmos or Geogebra.
There is an illustration of a version of the AM-GM inequality in the article Proof with pictures.
Possible extension
Inequalities is a short problem taken from a TMUA (Test of Mathematics for University Admission) question . Classical Means gives students the opportunity to prove the relationship between arithmetic, geometric and harmonic means using a diagram.