Weekly Challenge 1: Inner equality

7105 /www/nrich/html/content/id/7105/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <ul id="buttonBar"> <li> <a href="http://nrich.maths.org/4935&part=">Try this next</a> </li> <li> <a href="http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means">Read all about it</a> </li> <li> <a href="http://nrich.maths.org/5966&part=">Warm-up problem</a> </li> <li> <a href="http://nrich.maths.org/5986&part=">Last week's challenge</a> </li> </ul> 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? $$ ?? < a+ b- c - d < ?? $$ $$ ?? < a- c < ?? $$ $$ ?? < a - c + d - b < ?? $$ $$ ?? < abcd < ?? $$ $$ ?? < \frac{|a|+|c|}{2}-\sqrt{|ac|} < ??$$   <div class="framework"> 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.</div>     </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> 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 $$ 10 < a+ b- c - d < 20 $$ $$ -5 < a- c < 5 $$ $$ -10 < a - c + d - b < 10 $$ $$ 0 < abcd < 625 $$ $$ 0 < \frac{a+c}{2}-\sqrt{|ac|} < 2.5$$ 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. </mdoxml> 2 3 0 0 0 0 1 Weekly Challenge 1: Inner equality Our first weekly challenge. We kick off with a challenge concerning inequalities. Collections Weekly Challenge Algebra Inequality/inequalities Admin Stage 5 - Core Mapping Admin Stage 5 Decision mapping Stage 5 Core Mapping Document Polynomials AS Secondary Mapping Document DisplayCabinet