9.1 Workshop 1 (Exploration)

9.1.1 Preparatory exercises (45 mins)

Task 9.1.

Solve the following inequalities algebraically, with full justification, and sketch their corresponding region(s) on the Cartesian plane.

(a)

$x(x-1)<2$
(b)

$\frac{2}{x-1}<\frac{1}{x-2}$
(c)

$\frac{1}{x-1}<\frac{1}{x-2}$

Task 9.2.

Find the conditions on $a,b,c\in\mathbb{R}$ so that

(a)

$ax^{2}+bx+c<0$ for all $x\in\mathbb{R}$ .
(b)

$(a-b)^{2}+(b-c)^{2}+(c-a)^{2}>0$ .

Recall from Block 3 the following definition of the modulus (absolute value) of a real number.

Definition.

The modulus, or absolute value function, is defined by $|\cdot|:\mathbb{R}\to[0,\infty)$ where

|x|=\begin{cases}x,&\text{if }x\geq 0,\\ -x,&\text{if }x<0.\end{cases}

Task 9.3.

Solve the inequality

|x-1|<|x+2|

algebraically, with full justification, and sketch its corresponding region(s) on the Cartesian plane.

Task 9.4.

Find an expression for $f(n)$ and a value for $k\in\mathbb{N}$ that satisfy the following statements for all $n,N\in\mathbb{N}$ :

(a)

$\frac{2n}{n^{2}+4}<f(n)<\frac{2}{N}$ for all $n>N>k$ .
(b)

$\frac{1}{n^{2}-4}<f(n)<\frac{1}{N}$ for all $n>N>k$ .

9.1.2 Workshop tasks

Task 9.5 (Warm Up (5 mins)).

1.

Compare your solutions to Preparatory Tasks 9.1, 9.2 and 9.3 with the rest of your group. Do you all agree? Did you use similar or different methods?
2.

How does solving an inequality differ from solving an equation?

EXTENSION: Make a list of all the rules that the strict inequality $<$ obeys. Can you identify 4 core rules of the strict inequality $<$ from which all inequality rules for $<$ can be derived?

Task 9.6 (10 mins).

Find a sequence of bounds that prove the following inequalities hold by transitivity of inequalities.

1.

$\frac{n^{3}+2n^{2}-n}{n^{2}+n}\leq 3n$ for all $n\in\mathbb{N}$ .
2.

$\frac{n-1}{n^{2}+2n-1}\geq\frac{1}{4n}$ for all integers $n\geq 2$ .

EXTENSION: Suppose $k,t\in\mathbb{N}$ where $f(n)$ and $g(n)$ are polynomials of degree at most $k$ and $k+t$ respectively. Is it true that there exists some constant $c>0$ such that

\frac{n^{k}+f(n)}{n^{k+t}+g(n)}\leq\frac{c}{n^{t}}

for sufficiently large $n\in\mathbb{N}$ ?

Theorem (Triangle Inequalities).

For any real numbers $a,b\in\mathbb{R}$

1.

$|a+b|\leq|a|+|b|,$
2.

$|a-b|\geq||a|-|b||.$

Task 9.7 (10 mins).

Find a constant $k\in\mathbb{N}$ and a sequence of bounds that prove the following inequalities hold for all integers $n\geq k$ .

1.

$\left|\frac{n-3}{n^{2}+2n-4}\right|\leq\frac{1}{n}$
2.

$\left|\frac{5n^{2}+2n}{n^{2}+4}-5\right|\leqslant\frac{22}{n}$

EXTENSION: How would you prove the Triangle Inequalities?

Bernoulli’s inequality

Task 9.8 (10 mins).

Let’s start by convincing ourselves that the inequality is true for non-negative $x$ .

1.

Show that $(x+1)^{n}\geq 1+nx$ holds for all $x\geq 0$ when $n=3$ and $n=4$ .
2.

The Binomial Theorem states that, for all $x,y\in\mathbb{R}$ and $n\in\mathbb{N}$ ,

$(x+y)^{n}=\sum_{k=0}^{n}{\binom{n}{k}}x^{n-k}y^{k}\mbox{ where }\binom{n}{k}=% \frac{n!}{k!\,(n-k)!}.$

Use the Binomial Theorem to show that $(x+1)^{n}\geq 1+nx$ for all $x\geq 0$ and $n\in\mathbb{N}$ .

EXTENSION: Can you find examples of $n$ and $x$ for which this inequality does not hold?

Task 9.9 (15 mins).

Let’s look at the visual representation of this inequality.

1.

Sketch the graphs of $f(x)=(x+1)^{n}$ and $g(x)=1+nx$ for $n=2,3,4,5$ .
2.

How are $f(x)$ and $g(x)$ related? Can you verify your claim?
3.

Use your sketches to conjecture a larger range of $x$ for which the inequality

$(x+1)^{n}\geq 1+nx$

holds for all $n\in\mathbb{N}$ .
4.

Try and prove your conjecture (You may find induction helpful).

EXTENSION: For a specific value of $n$ , we may be able to extend our range of $x$ even further. Find the largest subset of $\mathbb{R}$ for which $(x+1)^{3}\geq 1+3x$ .

Cauchy–Schwarz inequality

Theorem (The Cauchy-Schwarz Inequality).

Given two sequences (lists) $a_{1},\,a_{2},\cdots,a_{n}$ and $b_{1},\,b_{2},\cdots,b_{n}$ of real numbers,

\sum_{k=1}^{n}a_{k}b_{k}\leq\sqrt{\sum_{k=1}^{n}a_{k}^{2}}\sqrt{\sum_{k=1}^{n}% b_{k}^{2}}.

Task 9.10 (10 mins - Optional Extension).

Use the Cauchy-Schwarz inequality to show that

x+y+z\leq 2\left(\frac{x^{2}}{y+z}+\frac{y^{2}}{x+z}+\frac{z^{2}}{x+y}\right),

for all $x,y,z>0$ .

Hint: Note that $\frac{x}{\sqrt{y+x}}\sqrt{y+x}$ .

EXTENSION: Can you find conditions that make the Cauchy-Schwarz Inequality hold with equality?