Question

If \(y>0,\) is \(x\) less than \(0 ?\) (1) \(x y=16\) (2) \(x-y=6\)

Short answer

No, x must be greater than 0.

Step-by-step solution

Analyze Condition from Statement (1)

According to statement (1), we have the equation: \[ xy = 16 \] Since given \( y > 0 \), the product \( xy \) is positive. Therefore, for \( xy = 16 \) to hold, \( x \) must also be positive.

Analyze Condition from Statement (2)

According to statement (2), we have the equation: \[ x - y = 6 \] Since given \( y > 0 \), in order for \( x - y = 6 \), \( x \) must be greater than \( y \). This implies that \( x \) is also greater than 0.

Evaluate Both Statements Together

Combining both statements, \( xy = 16 \) confirms that \( x \) is positive and \( x - y = 6 \) shows that \( x \) is greater than \( y \) which reinforces that \( x > 0 \).

Key concepts

inequalities

Inequalities are crucial in algebra. They allow us to understand the relationship between two expressions. In this exercise, we address the inequality with the condition, given that \(y > 0\). This means \(y\) is a positive number.
When dealing with inequalities, it's essential to remember the basic rules:

• If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same.
• If you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.

In statement (1), \(xy = 16\) and given \(y > 0\), meaning \(x\) must also be positive (\(x > 0\)).
In statement (2), \(x - y = 6\) and since \(y > 0\), \(x\) must be greater than \(y\). Therefore, \(x\) must be greater than 0. This illustrates how inequalities help to identify the nature (positive or negative) of variables in an equation.

algebra

Understanding algebraic expressions is pivotal to solve the given problem. Let's break down the equations:

• Statement (1): \(xy = 16\)
• Statement (2): \(x - y = 6\)

In statement (1), \(xy = 16\), given that \(y > 0\), implies that \(x\) must also be positive to keep the product \(xy\) positive.
In statement (2), \(x - y = 6\) and since \(y > 0\), rearranging the equation to \(x = y + 6\) shows \(x > y\). Thus, as \(y > 0\), \(x\) must be positive as well.
By examining both statements together, we can comfortably conclude that \(x > 0\). This simple algebraic manipulation and understanding the relationship between \(x\) and \(y\) ensures we solve the problem efficiently.

conditional analysis

Conditional analysis involves evaluating statements under given conditions. Here, the condition is \(y > 0\). This condition restricts \(y\) to be positive, which simplifies our analysis.
In statement (1), \(xy = 16\), knowing \(y > 0\) means \(x\) must be positive to satisfy the condition.
In statement (2), \(x - y = 6\), rearranging it to \(x = y + 6\) and using the condition \(y > 0\), we see that \(x\) must be positive.
When analyzing the problem conditionally, we combine the implications from both statements to reach a logical conclusion.
This kind of analysis is vital for problems on exams like the GMAT, where conditions guide us towards the solution by limiting the scope of variables.