How can I prove the statement |a-b| < |a| + |b| using the triangle inequality?

Therefore, |a-b| < |a| + |b|. In summary, the conversation is about proving the inequality |a-b| < |a| + |b| using the triangle inequality and the definition of absolute value. The individual is attempting to create formal proofs for different cases, but is struggling to make progress. However, it is noted that the inequality can be rewritten as |a-b| = |a+(-b)|, which can then be solved using the triangle inequality to show that the distance between a and b is smaller than the sum of the distances between a and 0 and 0 and b.
  • #1
Seda
71
0

Homework Statement



a and b are real numbers.

Show l a-b l < l a l + l b l



Homework Equations



Well, I know la+bl < lal + lbl by the triangle inequality.

The Attempt at a Solution



If I can prove that la-bl < la+bl, then I'm done, but that most recent inequality almost seems too intuitive to write a formal proof. I can use the definition of absolute value to create cases perhaps, but I always get lost and seem to go nowhere

Just to write the obvious stuff down.

la-bl = a-b if a-b> 0 (or a>b)
la-bl = b-a if a-b< 0 (or b>a)

la+bl = a+b if a+b>0 (or a>-b) .
la+bl = -a-b if a+b<0 ( or a<-b)

I can't seem how to go somewhere with this.
 
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  • #2
|a-b|=|a+(-b)|. Now you can use the triangle inequality.
 
  • #3
By triangle ineq., distance between a and b < distance between a and 0 + distance between 0 and b.
 

FAQ: How can I prove the statement |a-b| < |a| + |b| using the triangle inequality?

1. What is an absolute value proof?

An absolute value proof is a type of mathematical proof that involves showing the equality of two expressions involving absolute values. It is used to prove statements involving absolute value equations or inequalities.

2. How do you prove an absolute value expression?

To prove an absolute value expression, you must show that both the positive and negative versions of the expression are equal. This can be done by splitting the expression into two cases - one where the absolute value is positive and one where it is negative - and then solving for each case separately.

3. Can absolute value proofs be used for any type of equation or inequality?

Yes, absolute value proofs can be used for both equations and inequalities. However, they are most commonly used for equations.

4. Are there any specific strategies or techniques for solving absolute value proofs?

Yes, there are several strategies and techniques that can be used to solve absolute value proofs. These include splitting the expression into two cases, using algebraic manipulations to simplify the expression, and using properties of absolute values such as the triangle inequality.

5. How can absolute value proofs be applied in real-world situations?

Absolute value proofs can be applied in real-world situations where there is uncertainty or variability, such as in physics, economics, or statistics. They can also be used to solve problems involving distance, speed, and time.

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