Absolute Value Prove

[nycmath]

Precalculus
David Cohen
Edition 3
Chapter 1, Section 1.2

66. Show that for all real numbers a and b, we have | a | - | b | ≤ | a - b |

[Eigenvalue]

Rewrite |a-b| as |a-b+b-b|
|a-b|=|a-(b-b)|=|(a-b)+b|
The triangle inequality states that |x+y|[tex]\le[/tex]|x|+|y|
Let x=a-b and y=b
|(a-b)+b|[tex]\le[/tex]|a-b|+|b|
|a|[tex]\le[/tex]|a-b|+|b|
|a|-|b|[tex]\le[/tex]|a-b|

[nycmath]

Rewrite |a-b| as |a-b+b-b|
|a-b|=|a-(b-b)|=|(a-b)+b|
The triangle inequality states that |x+y|[tex]\le[/tex]|x|+|y|
Let x=a-b and y=b
|(a-b)+b|[tex]\le[/tex]|a-b|+|b|
|a|[tex]\le[/tex]|a-b|+|b|
|a|-|b|[tex]\le[/tex]|a-b|

A. Can you explain each step?

B. Can we replace a, b, and c with integers to make the math easier to grasp?

[Eigenvalue]

Let a=2, b=3
|a| can be rewritten as |a-b+b|, as adding b and subtracting b does not change the expression
In this case, |2| can be rewritten as |2-3+3|
The triangle inequality states that one side of the triangle is smaller than the sum of the other 2 sides: |x+y|[tex]\le[/tex]|x|+|y|
Let x=a-b and y=b
|x+y| can be replaced with |a-b+b| (or |a|), |x| is replaced with |a-b|, and y is replaced with |b|
|a|[tex]\le[/tex]|a-b|+|b|
|a|-|b|[tex]\le[/tex]|a-b|

We can try this with numbers:
|2|[tex]\le[/tex]|2-3|+|3|
2[tex]\le[/tex]1+3
2[tex]\le[/tex]4
The statement holds true

In hindsight, the first step of my previous solution was not needed

[nycmath]

Let a=2, b=3
|a| can be rewritten as |a-b+b|, as adding b and subtracting b does not change the expression
In this case, |2| can be rewritten as |2-3+3|
The triangle inequality states that one side of the triangle is smaller than the sum of the other 2 sides: |x+y|[tex]\le[/tex]|x|+|y|
Let x=a-b and y=b
|x+y| can be replaced with |a-b+b| (or |a|), |x| is replaced with |a-b|, and y is replaced with |b|
|a|[tex]\le[/tex]|a-b|+|b|
|a|-|b|[tex]\le[/tex]|a-b|

We can try this with numbers:
|2|[tex]\le[/tex]|2-3|+|3|
2[tex]\le[/tex]1+3
2[tex]\le[/tex]4
The statement holds true

In hindsight, the first step of my previous solution was not needed

Thank you so much.