If \(1 / y+1 / x<1 / 2\) : A. \(x y

Inequalities are mathematical expressions used to show that one quantity is larger or smaller than another. Instead of having an equal sign like an equation, inequalities use symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
Understanding inequalities is crucial because they are often used to describe ranges of possible values. For example, the inequality \(\frac{1}{y} + \frac{1}{x}< \frac{1}{2}\) tells us that the combined fractions of \(\frac{1}{y}\) and \(\frac{1}{x}\) must be less than \(\frac{1}{2}\).
Unlike equations, solving inequalities can involve flipping the inequality sign if we multiply or divide by a negative number. So always be cautious during those steps.

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. In this exercise, we work with variables \(x\) and \(y\) to manipulate an inequality. Basic algebraic operations include addition, subtraction, multiplication, and division.
One key strategy in algebra is to make an equation or expression simpler. Take the given inequality \(\frac{1}{y} + \frac{1}{x}< \frac{1}{2}\) for example. Combining the fractions by finding a common denominator, i.e., \(xy\), gives \(\frac{x + y}{xy} < \frac{1}{2}\).
Next, we eliminate the fraction by multiplying both sides by \(2xy\), transforming it into \[2(x + y) < xy \]. Finally, we rearrange it to \[xy - 2x - 2y > 0 \], which is simpler to compare with the given answer choices.

The Graduate Management Admission Test (GMAT) is a standardized test required for admission into many graduate business programs. It measures various skills including quantitative reasoning, verbal reasoning, and analytical writing.
Problems involving inequalities, like the one in this exercise, are common on the GMAT. These problems test not only your understanding of algebra but also your ability to simplify and manipulate complex expressions.
When preparing for the GMAT, practice solving different types of inequalities. Focus on learning how to simplify them and understand their graphical interpretations. Practicing under timed conditions can also help, as GMAT problems often need to be solved quickly and accurately.