Inequality of positive real numbers

In summary, inequality of positive real numbers is a mathematical concept denoted by the symbol ">" that describes the relationship between two or more numbers where one is greater than the other. It differs from equality in that it allows for a range of values rather than a specific value. It can be represented graphically on a number line and is used in various real-life situations such as comparing prices and calculating measurements. There are also several rules and properties associated with inequality of positive real numbers that allow for the manipulation of equations and inequalities involving multiple numbers.
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anemone
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If $x$ and $y$ are positive real numbers, prove that $4x^4+4y^3+5x^2+y+1\ge 12xy$.
 
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Applying the arithmetic-mean geometric-mean inequality three* times:

From $4x^4 + 1 \geq^* 2 \sqrt{4x^4} = 4x^2$ and $4y^3+y \geq^* 2\sqrt{4y^4}= 4y^2$ - we have

$4x^4+4y^3+5x^2+y+1 \geq 9x^2 + 4y^2 \geq^* 2\sqrt{36x^2y^2} = 12xy$.
 

Related to Inequality of positive real numbers

What is inequality of positive real numbers?

Inequality of positive real numbers refers to the relationship between two or more positive real numbers where one number is greater than, less than, or not equal to another number. This is denoted by the symbols > (greater than), < (less than), or ≠ (not equal to).

How is inequality of positive real numbers represented mathematically?

Inequality of positive real numbers is represented using mathematical symbols and equations. For example, the inequality 5 > 3 means that 5 is greater than 3. Similarly, the inequality 7 < 10 means that 7 is less than 10. Inequalities can also be represented using variables, such as x > 2 or y < 9.

What is the difference between an inequality and an equation?

An inequality compares two numbers or expressions and shows their relationship, such as greater than or less than. An equation, on the other hand, shows that two expressions are equal. Inequalities can have multiple solutions, while equations typically have one solution.

How is inequality of positive real numbers used in real life?

Inequality of positive real numbers is used in many real-life situations, such as comparing prices of items, determining the height or weight of objects, and calculating interest rates. It is also used in statistics to compare data and make conclusions about populations.

What are some common misconceptions about inequality of positive real numbers?

One common misconception is that the symbol > always means "greater than" and the symbol < always means "less than." However, these symbols can also represent "greater than or equal to" (≥) and "less than or equal to" (≤). Another misconception is that inequalities only involve numbers, but they can also involve variables and expressions. Lastly, it is important to note that inequalities do not always have a single solution, as there can be an infinite number of values that satisfy the inequality.

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