Inequality involving positive real numbers

In summary, "inequality involving positive real numbers" refers to mathematical expressions or equations that compare positive real numbers using symbols such as <, >, ≤, or ≥. It is important because it allows us to compare and analyze quantities in the real world and is used in various fields such as economics, physics, and social sciences. The basic rules include combining like terms, isolating the variable, and checking the solution. Real-world applications include analyzing income distribution, describing relationships between physical quantities, studying social inequalities, and making everyday decisions.
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anemone
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Prove that $\dfrac{y^2z}{x}+y^2+z\ge\dfrac{9y^2z}{x+y^2+z}$ for all positive real numbers $x,\,y$ and $z$.
 
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I use the fact, that the arithmetic mean is always greater than or equal to the harmonic mean (for positive reals):

\[\frac{\frac{y^2z}{x}+y^2 +z}{3} \geq \frac{3}{\frac{x}{y^2z}+\frac{1}{y^2}+\frac{1}{z}}\]

It follows immediately, that

\[\frac{y^2z}{x}+y^2 +z \geq \frac{9y^2z}{x+z+y^2}\]

q.e.d.
 

FAQ: Inequality involving positive real numbers

1. What is inequality involving positive real numbers?

Inequality involving positive real numbers refers to mathematical statements that compare two or more positive real numbers using the symbols <, >, ≤, or ≥. These symbols indicate whether one number is smaller or larger than another, or if they are equal.

2. How is inequality involving positive real numbers different from regular inequality?

Inequality involving positive real numbers specifically deals with numbers that are greater than zero. This is important because it allows for a more precise comparison between numbers, as it eliminates the possibility of negative numbers skewing the results.

3. What is the purpose of studying inequality involving positive real numbers?

Studying inequality involving positive real numbers allows us to make comparisons and draw conclusions about different quantities or values. It is also a fundamental concept in mathematics and is used in various fields such as economics, physics, and statistics.

4. What are some common methods for solving inequality involving positive real numbers?

There are several methods for solving inequality involving positive real numbers, including using algebraic manipulation, graphing on a number line, and using properties of inequalities (such as multiplying or dividing by a positive number).

5. How is inequality involving positive real numbers used in real-life situations?

Inequality involving positive real numbers is used in various real-life situations, such as determining the minimum or maximum amount of a product that can be produced given limited resources, analyzing income distribution in economics, and calculating probabilities in statistics.

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