# Unit Interval

Take any two numbers between 0 and 1. Prove that the sum of the
numbers is always less than one plus their product?

## Problem

Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product.

That is, if $0< x< 1$ and $0< y< 1$ then prove

$$x+y< 1+xy$$

## Getting Started

If you know $0< x< 1$ what can you say about $1-x$?

What can you say about the product of two positive numbers?

## Student Solutions

Given any two numbers between $0$ and $1$ you have to prove that their sum is less than 1 plus their product; that is, given $0 < x < 1$ and $0 < y < 1$, prove that $x + y < 1 + xy$.

Hyeyoun Chung, St Paul's Girls' School, and Andaleeb Ahmed, Woodhouse Sixth Form College, London both produced nice solutions.

Consider $1-x$ and $1-y$. Since $0 < x < 1$ and $0 < y < 1$ it follows that

$\begin{eqnarray} \\ (1 - x)(1 - y) & > & 0 \\ 1 - x - y + xy & > & 0 \\ 1 + xy & > & x + y. \end{eqnarray}$

This is equivalent to $x + y < 1 + xy$.

## Teachers' Resources

Why do this problem?

The problem gives practice in solving linear and quadratic inequalities.

Possible approach

Use this as a lesson starter. If learners do not know how to start let them use the Hint.

Then discuss the problem as a class but try to elicit ideas from the learners themselves.

Key questions

How do we make use of the information given?

If we are not making progress, then have we used all the information given?