Prove Inequality: llxl - lyll < lx - yl

In summary, the conversation discusses how to prove the inequality llxl - lyll < (or equal to) lx - yl for real numbers x and y, with suggestions to consider cases where both x and y are less than zero and the use of the triangle inequality. The concept of absolute value is also discussed, with explanations on its relationship to distance and its interpretation in terms of inequalities.
  • #1
Fairy111
73
0

Homework Statement


For real numbers x and y prove the following:

llxl - lyll < (or equal to) lx - yl


Homework Equations





The Attempt at a Solution



Im not really sure where to start, i was considering cases where x < 0 and say y< 0 and what that would imply say x-y would be. But I am not sure how to continue.
 
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  • #2
Hint: What is [itex](|x|-|y|)^2[/itex]?...How about [itex](|x-y|)^2[/itex]?
 
  • #3
Fairy111 said:

Homework Statement


For real numbers x and y prove the following:

llxl - lyll < (or equal to) lx - yl
.

You may need to consider a couple cases. There may be a little trick to doing this one.
I would say look at | x | equals and then look at what | y | equals.
So maybe add an subtract some thing to x, do the same for y. Then the triangle inequality says that
| a + b | <= |a| + |b| . This should be helpful in solving this. Does that help?
 
  • #4
ok, thankyou - i will have ago, althought I am not very good at proving things! Also what does the double modulus sign mean?
 
  • #5
Fairy111 said:
ok, thankyou - i will have ago, althought I am not very good at proving things! Also what does the double modulus sign mean?

Me either, I think it just takes a lot of patience. I may be misunderstanding your question, but I believe the fact that there is this nested absolute value in the inequality is done to provide a relationship that you will find useful, especially with sequences. I remember a couple of proofs that I would not have gotten without knowing the statement you are trying to prove.

Conceptually though it makes sense. Think about the modulus (absolute value) sign as a measure. Then what the statement below is saying something obvious. That if you have take the absolute value of some number A it will always be positive, likewise for B. So | A | - | B | is guaranteed to be less than A. On the other hand A - B is not guaranteed to be less than A, suppose A is positive and B is negative. And so the distance between | |A| - |B| | <= | A - B|. Or at least, that is how I conceptualize absolute values. There is also an inequality way to interprer them. Let k be some positive number. Then | x | <= k iff
-k <= x <= k
 

1. How do you prove the inequality llxl - lyll < lx - yl?

To prove this inequality, we need to use the properties of absolute value and basic algebraic manipulations. First, we can use the identity |a - b| = |b - a| to rewrite the left side of the inequality as |lx - ly|, and the right side as |x - y|. Then, we can use the triangle inequality property, which states that |a + b| ≤ |a| + |b|, to rewrite the left side as |lx - ly| ≤ |lx| + |ly|. Finally, we can substitute the values of lx and ly back into the inequality, and we will see that |lx| + |ly| = lx + ly, which is equivalent to the right side of the original inequality. Therefore, we have proven that llxl - lyll < lx - yl.

2. Why is it important to prove inequalities in mathematics?

Proving inequalities is important because it allows us to establish the relationships between different quantities and to make accurate comparisons. Inequalities are used in many areas of mathematics, such as calculus, linear algebra, and statistics, to name a few. By proving inequalities, we can also gain a deeper understanding of mathematical concepts and develop critical thinking skills.

3. Can you provide an example of a real-life situation where the inequality llxl - lyll < lx - yl is applicable?

Yes, this inequality can be applied in many real-life situations. For example, it can be used to compare the distances traveled by two objects moving at different speeds. Let's say object A travels at a speed of 50 mph and object B travels at a speed of 60 mph. If we want to compare the distances traveled by these two objects in a certain amount of time, we can use the inequality ll50t - 60tll < 50t - 60t, where t represents time in hours.

4. What are some common mistakes to avoid when proving inequalities?

Some common mistakes to avoid when proving inequalities include forgetting to consider the direction of the inequality (whether it is <, ≤, >, or ≥), incorrectly applying properties of absolute value, and making assumptions about the values of the variables. It is important to carefully follow the steps and use the correct properties and rules to ensure a valid proof.

5. How can I improve my skills in proving inequalities?

To improve your skills in proving inequalities, practice is key. Start by reviewing the basic properties and rules of inequalities, and then try solving different types of problems. It is also helpful to read and understand proofs written by others, and to seek feedback and guidance from a knowledgeable teacher or mentor. With persistence and practice, you can improve your skills and become proficient in proving inequalities.

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