When does the triangle inequality hold for absolute value?

- x - y - z <= x + y + z(by adding up the 2 inequalities)- 2 (x + y + z) <= 2 (|x| + |y| + |z|)(by dividing by 2)- (x + y + z) <= (|x| + |y| + |z|)(by multiplying both sides by -1)x + y + z >= -(x + y + z) >= -(|x| + |y| + |z|)(by definition of abs(x))x + y + z >= -x - y + -z >= -(x + y + z) >= -x - y - z(by definition of
  • #1
lepton123
11
0

Homework Statement


abs(x+y+z)≤abs(x)+abs(y)+abs(z) indicate when this equality holds and prove this statement


Homework Equations



Triangle inequality?

The Attempt at a Solution


I have nothing :/
 
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  • #2
What is the absolute value of x? if x => 0 then abs(x)=x, else it is -x.

So make a table with all of the possible cases and see what happens!
 
  • #3
Drat, I was hoping that that I wouldn't have to do a case analysis; is there a more elegant way of solving this though?
 
  • #4
I don't know elegant ... I grew up on a farm!

So once you have carried out the detailed work you can apply your own standards of elegance and cleverness ... and write something elegant!
 
  • #5
Assume (by renaming of variables) that x ≤ y ≤ z. Then you have 4 cases to check, it shouldn't be too tedious.
 
  • #6
x + y + z <= |x| + |y| + |z|
- x - y - z <= |x| + |y| + |z|

(definition of the absolute value)
 

What is absolute value equality?

Absolute value equality is a mathematical concept that refers to the condition where two quantities have the same absolute value, or magnitude, but may have different signs. In other words, it is the state of being equal in distance from zero on a number line.

How is absolute value equality different from regular equality?

In regular equality, two quantities must have the same value in order to be considered equal. However, in absolute value equality, the two quantities must have the same absolute value, regardless of their actual numerical value.

What are the rules for solving absolute value equations?

The rules for solving absolute value equations are as follows:1. Isolate the absolute value expression on one side of the equation.2. Set up two separate equations, one with the positive value of the absolute value expression and one with the negative value.3. Solve both equations for the variable.4. Check your solutions by plugging them back into the original equation.

What are some real-life applications of absolute value equality?

Absolute value equality can be applied in various fields, such as physics, engineering, and economics. For example, in physics, it is used to calculate the distance between two points or the difference between two forces. In engineering, it can be used to determine the tolerance level of a certain measurement. In economics, it can be used to calculate profits and losses.

How can I graph absolute value equations?

To graph an absolute value equation, plot the vertex (the point where the absolute value expression equals zero) and another point on either side of the vertex. Then, draw a V-shaped graph connecting the two points. Remember to label the axis and include the absolute value symbol on the graph.

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