Question

If \(x^{2}>y^{2}\) which statement must be correct? (A) \(x>y\) (B) \(xy^{3}\)

Short answer

The correct statement is (C) \(x \neq y\).

Step-by-step solution

Understand the Condition

The condition given is \(x^{2} > y^{2}\). This means the square of \(x\) is greater than the square of \(y\). It doesn't immediately tell us anything about \(x\) and \(y\) individually other than how their squares compare.

Consider Positive and Negative Values

If both \(x\) and \(y\) are positive, \(x^{2} > y^{2}\) implies \(|x| > |y|\), and since \(x, y\) are both positive, this simplifies to \(x > y\). If \(x\) and \(y\) are negative, the same condition \(x^{2} > y^{2}\) also implies \(|x| > |y|\), but \(x < y\). Therefore, \(x\) can be greater than \(y\) or less depending on the sign.

Analyze the Statement \(x \neq y\)

Given \(x^{2} > y^{2}\), if \(x\) were equal to \(y\), their squares would also be equal, which contradicts \(x^{2} > y^{2}\). Therefore, \(x\) must not be equal to \(y\), i.e., \(x eq y\).

Analyze the Statement \(x^3 > y^3\)

If \(x\) is positive and larger than \(y\), \(x^3 > y^3\). However, if both \(x\) and \(y\) are negative, \(x < y\) and \(x^3 < y^3\). Thus, \(x^3 > y^3\) is not always true.

Evaluate all Options

Options (A) \(x > y\) and (B) \(x < y\) depend on the sign of \(x\) and \(y\). Option (C) \(x eq y\) is universally true under the given condition. Option (D) \(x^3 > y^3\) is only true under certain conditions, but not always, hence it is not a "must be" correct statement.

Key concepts

Understanding Inequalities

Inequalities are a crucial part of PSAT Mathematics, and they represent a comparison between two values or expressions. An inequality indicates if one side is greater than, less than, or not equal to the other side. In our exercise, the inequality is expressed as \(x^2 > y^2\), meaning the square of \(x\) is greater than the square of \(y\). This form of inequality implies a relationship between the squared quantities, but it doesn’t give direct information about how \(x\) and \(y\) compare themselves.

When interpreting inequalities, it's essential to consider all possible scenarios, especially when dealing with squares, as both positive and negative numbers yield positive squares. Therefore, values of \(x\) and \(y\) could be either positive or negative, affecting how we interpret the inequality in practical problems. For example:

• If \(x = 3\) and \(y = -2\), then \(x^2 = 9\) and \(y^2 = 4\), making \(x^2 > y^2\) true, but \(x\) and \(y\) have opposite signs.
• If \(x = -4\) and \(y = -3\), \(x^2 = 16\) exceeds \(y^2 = 9\), though \(x < y\).

Exploring different possibilities helps in understanding the nature and restrictions of inequalities in the context of algebra.

Explaining Absolute Value

Absolute value is often tied to comparisons and inequalities because it represents the distance of a number from zero on the number line, without regard to direction. In mathematical notation, the absolute value of \(x\) is represented as \(|x|\).

Consider the condition \(x^2 > y^2\). It can imply that the magnitude of \(x\) is greater than \(y\), which can be rewritten in terms of absolute values as \(|x| > |y|\). This absolute comparison tells us how large the numbers are, but not whether \(x\) itself is greater or smaller than \(y\).

A deeper look can show:

• If both \(x\) and \(y\) are positive, then simply comparing the absolutes gives us: \(x > y\).
• If both are negative, the inequality \(|x| > |y|\) means \(x\) is a larger value negatively (closer to zero), hence \(x < y\).

It's crucial when solving inequalities or absolute value problems to remember the properties of the number line and how positive and negative values are treated in absolute terms.

Decoding Algebraic Expressions

Algebraic expressions form the backbone of equations and inequalities in mathematical problems. They are combinations of variables, numbers, and operations that describe a particular relationship or condition.

In our context, expressions like \(x^2 - y^2\) give a view of how inequalities can be analyzed. When an expression states \(x^2 > y^2\), it sets a scene where the terms \(x\) and \(y\) react differently based on their actual values.

One useful way to dissect such inequalities is by factoring. For instance, using the identity \(x^2 - y^2 = (x-y)(x+y)\), we see how the differences and sums of roots influence the solution. This identity shows that the differences in the squares are also governed by the sum and difference of the numbers themselves. Thus:

• If \(x > y\), the difference \(x-y>0\), aligning with the condition \(x^2 > y^2\).
• If \(x < y\), then \(x-y<0\), opposite to what is hypothesized initially, therefore not the case for \(x^2 > y^2\) in standard direction.

Such comprehension equips you to tackle similar problems in algebra efficiently, unraveling each part of the expressions to make sense of the criteria given.