How does the proof connect the principles to show the elementary inequality?

In summary, from the principles of absolute value and triangle inequality, it can be deduced that |a - b| < 1 implies |a| < |b| + 1. This can be shown by using the fact that |a| = |(a + b) - b| and applying the triangle inequality to |a - b|.
  • #1
duke_nemmerle
50
1
I'm looking over a proof and I'm wondering from which principles does it follow that
[tex] \mid a - b \mid < 1 \to \mid a \mid < \mid b \mid + 1 [/tex]

I can see that [tex] |a - b | \le |a| + |-b| = |a| + |b| [/tex] and that [tex] |a| - |b| < |a| + |b| [/tex] but I just can't connect the dots.
 
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  • #2
duke_nemmerle said:
I'm looking over a proof and I'm wondering from which principles does it follow that
[tex] \mid a - b \mid < 1 \to \mid a \mid < \mid b \mid + 1 [/tex]

I can see that [tex] |a - b | \le |a| + |-b| = |a| + |b| [/tex] and that [tex] |a| - |b| < |a| + |b| [/tex] but I just can't connect the dots.


Ahh, got it [tex] |a| = |(a+b)-b| \le |a-b| + |b|[/tex] which means [tex] |a| - |b| \le |a-b| [/tex] the result immediately follows
 

Related to How does the proof connect the principles to show the elementary inequality?

1. What is an elementary inequality?

An elementary inequality is a mathematical statement that compares two numbers or expressions using symbols such as <, >, ≤, or ≥. It is typically used to describe relationships between quantities and is an essential concept in algebra and calculus.

2. What are the different types of elementary inequalities?

There are several types of elementary inequalities, including linear inequalities, quadratic inequalities, rational inequalities, and absolute value inequalities. Each type has its own unique characteristics and methods for solving.

3. How are elementary inequalities solved?

Elementary inequalities can be solved algebraically by using basic arithmetic operations and properties, as well as algebraic techniques such as factoring and solving systems of equations. Graphical methods, such as plotting the inequality on a number line, can also be used to solve certain types of inequalities.

4. What is the importance of elementary inequalities in science?

Elementary inequalities are important in science because they allow us to describe and analyze relationships between variables and make predictions about the behavior of systems. In many scientific fields, such as physics and chemistry, inequalities are used to model and solve complex problems.

5. Can elementary inequalities be applied in real-world situations?

Yes, elementary inequalities can be applied in real-world situations. For example, they can be used to determine the maximum or minimum values of a function in optimization problems, or to analyze data and make conclusions in scientific experiments. They are also commonly used in economics, engineering, and other fields to model and solve practical problems.

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