# Absolute Value Proof (Need Help)

• Seda
In summary, this equation can be solved by taking the single number that is (a+b), setting x=a+b, y=c, and solving for d.
Seda

## Homework Statement

Prove that abs(a+b+c) is less than or equal to abs(a)+abs(b)+abs(c)

None

## The Attempt at a Solution

This makes sense to me that this would always be true, but i just can't seem to figure out how to write it out

Write it as abs((a+b)+c) then use the triangle inequality a couple times.

Are you asking to prove |a+b+c| < |a| + |b| + |c| given |a+b| < |a| + |b|? Or do you need to prove |a+b| < |a| + |b| as well?

Yes, we are given that the triangle inequality is true, (and we also know how to prove the triangle inequality if that helps.)

But it doesn't seem that I can prove this the same way.

I know abs(a+b) <= abs(a) + abs(b)

So I can very easily get that abs(a+b) + abs(c) <= abs(a) + abs(b) + abs(c)

But how do I a get the left half to what i want?

Let d = a+b. |d+c| < |d|+|c| by tri. ineq.

So, |a+b+c| < |a+b|+|c|.

Now use the tri. ineq. on a+b, then add |c| to both sides.

Seda said:
Yes, we are given that the triangle inequality is true, (and we also know how to prove the triangle inequality if that helps.)

But it doesn't seem that I can prove this the same way.

I know abs(a+b) <= abs(a) + abs(b)

So I can very easily get that abs(a+b) + abs(c) <= abs(a) + abs(b) + abs(c)

But how do I a get the left half to what i want?

Think of (a+b) as a single number. That is, look at |x+ y| and then set x= a+b, y= c.

Wow...it seems so obvious now...

Sometimes I feel like a genius, others I feel stupid as hell, guess what's the case now?

Thanks so much.

## 1. What is absolute value?

Absolute value is a mathematical concept used to measure the distance between a number and zero on a number line. It is always positive and represented by two vertical bars surrounding the number.

## 2. How do you prove absolute value?

To prove the absolute value of a number, you need to show that it is equal to the distance between the number and zero on the number line. This can be done by using the definition of absolute value and performing algebraic manipulations.

## 3. Why is absolute value important?

Absolute value is important in many mathematical and scientific fields because it allows us to evaluate the magnitude of a number without considering its direction. It is also used in solving equations and inequalities, as well as in geometric and statistical analysis.

## 4. Can you give an example of an absolute value proof?

Consider the absolute value of -5. Using the definition, we can say that |-5| = 5, which is equal to the distance between -5 and 0 on the number line. Therefore, we have proved that the absolute value of -5 is 5.

## 5. Are there any properties of absolute value that can help in proving equations?

Yes, there are several properties of absolute value that can be helpful in proofs. These include the triangle inequality, which states that the absolute value of the sum of two numbers is less than or equal to the sum of their absolute values, and the multiplication property, which states that the absolute value of a product is equal to the product of the absolute values of the factors.

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