SOLUTION: If a < b, show that a < (a + b)/2 < b. The number (a + b)/2 is called the arithmetic mean of a and b.
Algebra.Com
Question 1207706: If a < b, show that a < (a + b)/2 < b. The number (a + b)/2 is called the arithmetic mean of a and b.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
We want to show that a < (a + b)/2 < b
This breaks down into these two key parts
a < (a + b)/2 and (a + b)/2 < b
What I'll do is start with the first piece and follow these steps
a < (a + b)/2
2a < a + b
2a-a < a + b - a
a < b
Next, I'll reverse those steps by following the chain upward.
a < b
a+a < b+a
2a < a+b
a < (a+b)/2
We have gone from the given a < b to prove that a < (a+b)/2 is the case.
Follow similar steps for the other needed key component.
a < b
a+b < b+b
a+b < 2b
(a+b)/2 < b
We have shown that if a < b, then
a < (a + b)/2 and (a + b)/2 < b
in which those two pieces glue together to get
a < (a + b)/2 < b
--------------------------------------------------------------------------
Another approach
Let c = a+b
Add 'a' to both sides of the original inequality.
a < b
a+a < b+a
2a < a+b
2a < c
Return back to the original inequality.
This time add b to both sides.
a < b
a+b < b+b
c < 2b
Since 2a < c and c < 2b, we know that 2a < c < 2b.
Then,
2a < c < 2b
2a < a+b < 2b
2a/2 < (a+b)/2 < 2b/2
a < (a+b)/2 < b
RELATED QUESTIONS
If a < b, show that a < (a + b)/2 < b.
Note: The number (a + b)/2 is called the... (answered by Edwin McCravy)
If a < b, show that a < (a + b)/2 < b.
Note: The number (a + b)/2 is called the... (answered by ikleyn,math_tutor2020)
If 0 < a < b, show that a < sqrt{a•b} < b. The number sqrt{a•b} is called the... (answered by math_tutor2020)
if arithmetic mean and geometric mean are inserted between a and b, such that arithmetic (answered by Edwin McCravy)
For 0 < a < b, let h be defined by (1/h) = (1/2)[(1/a) + (1/b)].
Show that a < h < b.... (answered by Edwin McCravy,math_tutor2020,mccravyedwin)
For 0 < a < b, let h be defined by
1/h = (1/2)[(1/a) + (1/b)].
Show that a < h < b. (answered by Edwin McCravy,mccravyedwin,ikleyn)
If 0 < a < b, show that a < sqrt{ab} < b.
Note: The number sqrt{ab} is called the... (answered by Edwin McCravy)
Show that A = B, when A = {a | a is even number} and B = {b| b is multiple of... (answered by psbhowmick)
What is the arithmetic mean between a and... (answered by KMST)