When working with fractions and inequalities, a common task is to compare them or derive relationships from their values. The statement that if ( frac{a}{b}

Proof of the Inequality Transformation

To prove that if ( frac{a}{b} 0 ) and ( d > 0 ).

Step 1: Rewrite the Inequality Using Cross-Multiplication

Given the inequality ( frac{a}{b} [ a cdot d

This step is valid because multiplying both sides of an inequality by a positive number (in this case, ( b ) and ( d )) does not change the direction of the inequality. This is a fundamental property of inequalities, essential for our proof.

Step 2: Verify the Assumptions

The proof relies on the assumption that ( b ) and ( d ) are both positive. If either ( b ) or ( d ) were negative, the inequality would reverse. Therefore, we must explicitly state this condition to ensure the proof is valid:

This ensures that the cross-multiplication step maintains the inequality's direction.

Step 3: Address the Sign Considerations

It’s important to note that the sign of the denominators can impact the validity of the inequality after cross-multiplication. If both ( b ) and ( d ) are positive, the inequality holds as discussed. However, what happens if one or both denominators are negative? Let’s explore these cases:

Case 1: Both ( b ) and ( d ) are Positive

If both ( b ) and ( d ) are positive, the cross-multiplication rule ( frac{a}{b}

Case 2: Both ( b ) and ( d ) are Negative

If both ( b ) and ( d ) are negative, multiplying both sides of the inequality by ( bd ) (which is positive) will result in the same direction of the inequality:

Case 3: ( b ) and ( d ) Have Opposite Signs

When ( b ) and ( d ) have opposite signs, ( bd ) will be negative, and multiplying both sides of the inequality by ( bd ) will reverse the direction of the inequality:

This case demonstrates that the inequality direction changes if the denominators have opposite signs.

Conclusion

In summary, the relationship ( a cdot d

Note

This proof assumes that ( b eq 0 ) and ( d eq 0 ). If any of these are zero, the original fractions are undefined, and the inequality cannot be transformed as described in this proof.

Understanding these concepts is crucial for solving more complex algebraic problems and for applications in mathematics, physics, and engineering where inequalities between fractions or ratios are often used.