Problem 65 - Intermediate Algebra

Question

Can an inequality in two variables be an identity, one that is satisfied by all pairs \((x, y) ?\) If so, give an example.

Step-by-Step Solution

Answer

Yes, an example is the inequality \( x^2 + y^2 \geq 0 \).

Step 1: Understanding Inequality in Two Variables

An inequality in two variables involves expressions such as \( ax + by \leq c \). To determine if it can be an identity, it must satisfy all pairs \((x, y)\) regardless of their specific values.

Step 2: Exploring the Concept of Identity

An identity is a mathematical statement that holds true for all possible values of the variables involved. For instance, the equation \( 0 = 0 \) is an identity because it is always true.

Step 3: Utilizing the Zero Inequality

Consider the inequality \( 0 \leq 0 \). This is true for any pair \((x, y)\) because zero is always less than or equal to zero. Thus, any expression equivalent to zero on both sides forms an identity.

Step 4: Constructing an Identity with Inequality

To illustrate with an example: \( 0 \leq x^2 - x^2 \). Here, \( x^2 - x^2 = 0 \) for any real number \( x \), thereby satisfying the condition for every pair \((x, y)\). Similarly, \( x^2 + y^2 \geq 0 \) is an inequality that holds for all real \( x \) and \( y \).

Key Concepts

Identity in Inequalities

An identity in math is a statement that is always true for any variables chosen.
In the context of inequalities involving two variables, an identity would be an inequality that holds true no matter what values are assigned to the variables involved.
For example, suppose we have an inequality like \( ax + by \leq c \): if this is an identity, then it must be true for *all* possible \(x\) and \(y\) values.
This concept requires the inequality to hold universally.Identities are typically seen in equations, but in inequalities, they emerge when the inequality holds under every condition. Consider a simple example like \( 0 \leq 0 \): it seems straightforward because zero is always equal to zero.

• Such statements form the root of identifying identities in inequalities.

Zero Inequality

The zero inequality is a powerful concept in understanding identities within inequalities.
The inequality \( 0 \leq 0 \) is inherently true because zero is equal to itself and is not greater than itself. This makes \( 0 \leq 0 \) a trivial example of an inequality identity.When thinking of how this applies to other inequalities, imagine substituting zero in expressions:

• Take \( x^2 - x^2 \): no matter what \( x \) is, this simplifies to \( 0 \).
• The inequality then becomes \( 0 \leq 0 \), which is always true.
• Hence, it serves as an identity for all real \( x \).

This concept extends to inequalities in multiple variables as well. Consider \( x^2 + y^2 \geq 0 \), which states that the sum of two squares is always non-negative. Similar to the zero inequality, this condition holds for any real numbers \( x \) and \( y \), showcasing an identity in inequalities.

Expressions in Algebra

Expressions in algebra are the building blocks of mathematical equations and inequalities.
They consist of variables, numbers, and operations like addition, subtraction, multiplication, and division. In algebra, expressions can be as simple as \( x + y \) or as complex as \( ax^2 + by + c \).To construct identities within inequalities, one often simplifies expressions to see if they equate to a universally true statement. For example:

• If we have an expression like \( x^2 - x^2 \), it simplifies to just \( 0 \).
• Substituting this into an inequality yields \( 0 \leq 0 \), a universal truth.

These relationships simplify the verification process for inequalities in two variables, as you can reduce complex expressions to simpler forms that either hold true universally or for specified conditions.