Use the rules of inequalities to prove the following...

Use the rules of inequalities to prove the following inequalities. If $a, b, c,$ and $d$ are positive numbers such that $\frac{a}{b}<\frac{c}{d},$ then $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d} \cdot[$ Hint: Show that $\left.\frac{a d}{b}+a<c+a \text { and } a+c<\frac{c b}{d}+c .\right]$

Use the rules of inequalities to prove the following inequalities.
If $a, b, c,$ and $d$ are positive numbers such that $\frac{a}{b}<\frac{c}{d},$ then $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d} \cdot[$ Hint: Show that $\left.\frac{a d}{b}+a<c+a \text { and } a+c<\frac{c b}{d}+c .\right]$

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Key Concepts

Properties of Inequalities

This concept involves the fundamental rules that govern how inequalities can be manipulated and combined. Such rules include adding or subtracting the same quantity from both sides of an inequality, multiplying or dividing both sides by a positive number, and maintaining the order when combining inequalities. These properties are essential for transforming and proving new relations from given inequalities without changing their truth value.

Mediant Fraction

The mediant of two fractions, defined as the fraction whose numerator is the sum of the numerators and whose denominator is the sum of the denominators, lies strictly between the two original fractions provided the fractions are in their reduced form and in proper order. Understanding this concept allows one to establish bounds on ratios and is a common technique in proving inequalities involving fractions.

Algebraic Manipulation of Rational Expressions

This key concept covers the techniques employed in handling and simplifying expressions that involve fractions. It includes cross-multiplication, common denominator methods, and careful rearrangement of terms. Such manipulations are critical in proving inequalities by transforming complex fractional expressions into more manageable forms where the standard rules of inequalities can be applied directly.

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