91Ó°ÊÓ

Understanding the reciprocal of an inequality is crucial when dealing with fractional variables. In our case, we are given the inequality \( \frac{1}{x} > \frac{4}{3} \). To solve for \( x \), we need to consider the reciprocal of both sides. The reciprocal of a number \( a \) is \( \frac{1}{a} \) and vice versa.

When taking the reciprocal of both sides of an inequality, the direction of the inequality changes if both sides have the same sign. So, in our example, because both fractions were positive, after reciprocating, the sign flipped, leading to \( x < \frac{3}{4} \). Remember, if one side was negative, we would not flip the inequality sign upon reciprocating.

The concept of flipping the inequality sign is one that often puzzles students, but it is a straightforward rule once understood. The main scenario where an inequality sign flips is when you multiply or divide both sides of the inequality by a negative number. However, another situation where this can occur, which can be less intuitive, is when taking reciprocals, as seen in our previous section.

To clarify, whenever you take the reciprocal of a positive variable across an inequality, the sign flips, as was the case in our example, from \( > \) to \( < \). It is important to highlight that the reciprocal of a negative number does not result in a sign flip, because the number remains negative.

Simplifying inequalities involves reducing them to their simplest form to make a clear and direct comparison between the variables involved. In our example, it entailed isolating the variable and combining like terms. Initially, we have \( \frac{1}{x} - 1 > \frac{1}{3} \). To simplify, we first removed the constant term from one side by adding 1 across the inequality, yielding \( \frac{1}{x} > \frac{4}{3} \).

Being attentive to this progression is vital for solving inequalities thoroughly and accurately. By maintaining the properties of inequality and operating systematically, we can transform even complex inequalities into their simpler and more solvable forms, as exemplified by our concise final answer \( x < \frac{3}{4} \).