3 Inequalities

Question 1

Definition 3.1:
Absolute value

Answer 1

The absolute value |x| of x ∈ R is
* x if x ≥ 0,
* −x if x < 0.

Question 2

Proposition 3.1:
Properties of absolute values

Answer 2

For all x ∈ R,
* x ≤ |x|
* − x ≤ |x|
* |−x|=|x|

For all x,y ∈ R,
* |xy| = |x||y|

Question 3

Proposition 3.2:
Triangle inequality

Answer 3

For all x, y ∈ R,
|x + y| ≤ |x| + |y|

Question 4

Proposition 3.2:
Reverse triangle inequality

Answer 4

For all x, y ∈ R,
||x| − |y|| ≤ |x − y|

Question 5

Proposition 3.3:
Binomial inequality

Answer 5

For all x ∈ R and all n ∈ N, if x ≥ −1, then (1+x)ⁿ ≥1+nx.