Question

Is \(x^2>y^2 ?\) (1) \(x-y>0\) (2) \(x\) is negative A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Short answer

C. Statements (1) and (2) together are sufficient.

Step-by-step solution

- Analyze Statement (1)

Given condition: (1) \(x - y > 0\) Reframe it: \(x > y\). If \(x > y\), we know one variable is greater than the other. But since there are no constraints on whether those variables are positive or negative, we can't directly conclude \(x^2 > y^2\) from this statement alone.

- Analyze Statement (2)

Given condition: (2) \(x\) is negative. This means \(x < 0\). On its own, this also does not provide enough information about their relationship to determine if \(x^2 > y^2\).

- Combine Statements (1) and (2)

Combine the information from both statements: - From (1) \(x > y\) - From (2) \(x < 0\) (since \(x\) is negative) If both understandings are applied, \(x\) is negative and still greater than \(y\). Therefore, \(y\) must be even more negative than \(x\). As \(x\) is greater than \(y\) in the negative realm, the magnitude of \(y\) is bigger than the magnitude of \(x\). Since both are negative and \(|y|> |x|\), \(x^2 < y^2\).

- Conclusion

Given both statements, we prove \(x^2 < y^2\), which invalidates \(x^2 > y^2\). The combined information definitively answers the question.

Key concepts

inequalities

Inequalities help us understand relationships between numbers. An inequality like \(x > y\) shows that \(x\) is bigger than \(y\).
Inequalities are useful in problem-solving, especially when comparing variables. For example:

• \( x > y \) means \( x \) is bigger.
• \( x < y \) means \( x \) is smaller.

In our problem, knowing \( x - y > 0 \) or \( x > y \) tells us about the order of the variables. However, inequalities alone don't give us enough information to conclude about the squares of \( x \) and \( y \), especially without knowing their negativity or positivity.

negative numbers

Negative numbers are less than zero. When dealing with inequalities and negative numbers, it’s important to consider their special properties.
For example:

• The product of two negative numbers is positive (\( (-a) \times (-b) = ab \)).
• When comparing magnitudes, a number closer to zero is smaller.

In our exercise, knowing \( x \) is negative changes how we compare magnitudes. If \( x \) is a negative number, then \( x < 0 \).
This adds a layer of complexity, as `-5` is greater than `-10`, but `-10` has a larger magnitude.

magnitude comparison

Magnitude comparison checks the absolute sizes, ignoring signs.
For example, `|x|` means the absolute value of \( x \).

• \(|3| = |-3| = 3\)
• \(|-7| = 7\), so it’s larger than \(|-2| = 2\)

In our case, \( x \) is greater than \( y \) yet both are negative.
Hence, \( y \) is `more negative` making \( |y| > |x| \). In simpler terms, since both are negative and \( y \) has a larger absolute value, \( y \) has a smaller algebraic value. Therefore, when comparing squares, \( y^2 \) is larger.

algebra

Algebra helps to solve equations and inequalities by manipulating expressions.
It involves understanding variables like \(x\) and \(y\), and performing operations on them. Basic rules include:

• Combining like terms (\(2x + 3x = 5x\)).
• Applying distributive properties \(a(b + c) = ab + ac\).

In our problem, algebra helps us combine statements:
\(x - y > 0\) is the same as saying \(x > y\).
Even though \( x < 0 \), combining both tells us \( y \) is even more negative. Understanding this algebraic relationship is key to determining that \( x^2 < y^2 \).