91Ó°ÊÓ

Rational expressions are fractions where both the numerator and the denominator are polynomials. In the context of inequalities, like in the given exercise, rational expressions help us understand how quantities relate to each other. When comparing two rational expressions, it's crucial to consider the values of their numerators and denominators separately.
- **Numerators**: These are the values above the fraction line. For instance, in \( \frac{a}{d} \), "a" is the numerator.- **Denominators**: These values come after the fraction line, such as "d" in \( \frac{a}{d} \).
When analyzing inequalities with rational expressions, always remember to maintain the order of operations and the properties of inequalities, which will be explained more in the next section.

The properties of inequalities allow us to manipulate and compare different expressions effectively. Here are crucial properties that apply to solving inequalities involving rational expressions:
- **Multiplying or Dividing by a Positive Number**: If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality stays the same.- **Multiplying or Dividing by a Negative Number**: If you multiply or divide both sides of an inequality by a negative number, the direction of the inequality reverses.
In the original problem, we multiplied both sides of the inequality by \( cd \) because both "c" and "d" are positive numbers. This does not change the inequality's direction, maintaining the relationship between the expressions.
Always ensure the operations you perform on inequalities are valid and preserve the inequality's direction, especially when dealing with rational expressions.

A counterexample is a specific case that shows a statement is false. In mathematics, providing a counterexample can be as valid as proving a statement. In the exercise, if we assume that \( \frac{a}{d} < \frac{b}{c} \) is always true under the given conditions, a single counterexample could disprove this assumption.
To create a counterexample:

• Select specific values for \(a, b, c,\) and \(d\) such that the conditions \(0 < a < b\) and \(0 < c < d\) hold.
• Calculate both expressions \( \frac{a}{d} \) and \( \frac{b}{c} \).
• Check if \( \frac{a}{d} \geq \frac{b}{c} \). If it does, you have a valid counterexample.

Counterexamples help explain why universal claims may not hold by showing exceptions.

Comparing fractions can sometimes be tricky, but understanding their behavior is essential. In this exercise, the goal is to determine if one fraction is less than another given specific conditions.
To compare two fractions:- Find a common denominator if necessary, although it's not always needed.- Cross-multiply to compare non-similar denominators directly, as done in the original solution's step.Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other and vice-versa, which helps easily compare their values.For example, to check if \( \frac{a}{d} < \frac{b}{c} \):- Multiply "a" by "c" and "b" by "d" to get \( ac \text{ and } bd \).- Compare these products directly: if \( ac < bd \), then \( \frac{a}{d} < \frac{b}{c} \).
This method is often simpler than converting fractions to decimals, especially when dealing with inequalities.