Prove y is less than or equal to z

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In summary, proving that y is less than or equal to z involves using evidence or logical reasoning to show that y is either smaller than z or equal to z. This can be done through various mathematical techniques such as algebraic manipulation or using logical statements and proofs. It is important to prove y is less than or equal to z in order to validate mathematical statements and make accurate conclusions. However, some common mistakes when attempting to prove this include forgetting necessary steps, using incorrect mathematical operations, or assuming the inequality without proof. It is crucial to double-check the proof and ensure all steps are accurate.
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anemone
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Let $x,\,y,\,z,\,a,\,b$ be positive real such that $xa+yb\le ya+zb\le za+xb$.

Prove that $y\le z$.
 
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  • #2
anemone said:
Let $x,\,y,\,z,\,a,\,b$ be positive real such that $xa+yb\le ya+zb\le za+xb$.

Prove that $y\le z$.
if $y>z------(1)$
we have:$ay+bz\leq az+bx<ay+bx$
$\therefore bz<bx,\, or \,,x>z----(2)$
$az+bx\geq ax+by>az+by$
$\therefore bx>by ,\,\, or \,\, x>y---(3)$
from (1)(2)(3)we get :$x>y>z$
this contradicts to $ax+by\leq ay+bz$
so we obtain $y\leq z$
it is easy to see $y=z=x$ is possible
 
  • #3
Thanks for participating, Albert!

Solution of other:

The first inequality tells us:

\(\displaystyle \color{yellow}\bbox[5px,purple]{xa+yb\le ya+zb}\color{black}\le za+xb\)

\(\displaystyle (y-z)b\le a(y-x)\)

Whereas the second inequality is equivalent to

\(\displaystyle \color{black}xa+yb\le\color{yellow}\bbox[5px,green]{ ya+zb\le za+xb}\)

\(\displaystyle (y-z)a\le b(x-z)\)

If $y>z$, then the left sides of the two inequalities above are positive, so the right sides are positive as well.

In particular, this means $y-x>0$ and $x-z>0$, this further implies $y>x>z$, giving

$xa>za$ and $yb>xb$

Adding them up gives $xa+yb>za+xb$, which contradicts the relation between the first and third expressions in the given original inequality.

Thus, $y\le z$.
 

FAQ: Prove y is less than or equal to z

1. What does it mean to "prove y is less than or equal to z"?

Proving that y is less than or equal to z means providing evidence or logical reasoning to demonstrate that y is either smaller than z or equal to z.

2. How do you prove y is less than or equal to z?

To prove y is less than or equal to z, you can use various mathematical techniques such as algebraic manipulation, comparison of numerical values, or using logical statements and proofs.

3. Can you provide an example of a proof for y is less than or equal to z?

Sure, for example, if we have the inequality 2y + 4 ≤ 2z, we can subtract 4 from both sides to get 2y ≤ 2z - 4. Then, we can divide both sides by 2 to get y ≤ z - 2. This shows that y is less than or equal to z, using algebraic manipulation.

4. Why is it important to prove y is less than or equal to z?

Proving y is less than or equal to z is important because it helps validate mathematical statements and ensures that they are accurate. Additionally, it allows us to make conclusions and solve problems based on the given inequality.

5. What are some common mistakes when attempting to prove y is less than or equal to z?

Some common mistakes when proving y is less than or equal to z include forgetting to include all necessary steps in the proof, using incorrect mathematical operations, or assuming the inequality without proving it. It is important to double-check your work and make sure all steps are clear and accurate in your proof.

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