# Inequality involving a, b, c and d

• MHB
• anemone
In summary, inequality involving a, b, c, and d is a mathematical expression that compares the values of these variables and shows that one value is greater than, less than, or not equal to another value. To solve an inequality involving a, b, c, and d, you must follow the same rules as you would when solving an equation, but with additional rules such as changing the direction of the inequality sign when multiplying or dividing by a negative number. An inequality compares the values of two expressions, while an equation shows that two expressions are equal to each other. Inequalities use symbols like <, >, ≤, ≥, and ≠, while equations use the equal sign (=). An inequality can have an infinite number of solutions
anemone
Gold Member
MHB
POTW Director
Given the real numbers $a,\,b,\,c$ and $d$, prove that

$(1+ab)^2+(1+cd)^2+a^2c^2+b^2d^2\ge 1$

Expanding the LHS of the inequality we get $1+2ab+a^2b^2+1+2cd+c^2d^2+a^2c^2+b^2d^2=1+(1+ab+cd)^2+(ac-bd)^2\ge 1$

## 1. What is the definition of inequality involving a, b, c and d?

An inequality involving a, b, c, and d is a mathematical statement that compares the values of these variables using inequality symbols such as <, >, ≤, or ≥.

## 2. How do you solve inequalities involving a, b, c and d?

To solve an inequality involving a, b, c, and d, you must isolate the variable on one side of the inequality symbol and then use the properties of inequality to determine the possible values for that variable.

## 3. What are the different types of inequalities involving a, b, c and d?

There are three main types of inequalities involving a, b, c, and d: linear inequalities, quadratic inequalities, and systems of inequalities. Linear inequalities involve variables raised to the first power, quadratic inequalities involve variables raised to the second power, and systems of inequalities involve multiple inequalities with multiple variables.

## 4. How do inequalities involving a, b, c and d relate to real-life situations?

Inequalities involving a, b, c, and d can be used to represent real-life situations such as income inequality, where a, b, c, and d can represent different levels of income for different individuals. They can also be used in business and economics to represent supply and demand, production costs, and profit margins.

## 5. What are some common mistakes to avoid when solving inequalities involving a, b, c and d?

Some common mistakes to avoid when solving inequalities involving a, b, c, and d include forgetting to switch the direction of the inequality symbol when multiplying or dividing by a negative number, not properly distributing when simplifying expressions, and not checking the solution in the original inequality to ensure it is valid.

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