Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.8 - Inequalities - 1.8 Exercises - Page 91: 130

Answer

$\dfrac{a}{b}\lt \dfrac{a+c}{b+d}\lt \dfrac{c}{d}$

Work Step by Step

Consider the inequality $\dfrac{a}{b}\lt \dfrac{c}{d}$ This gives: $\dfrac{ad}{b}\lt c$ ....(1) Also, we have $\dfrac{a}{b}\lt \dfrac{c}{d}$ $a\lt \dfrac{bc}{d}$ or, $a+c\lt \dfrac{bc}{d}+c$ ...(2) Combine the both equations (1) and (2). This gives: $\dfrac{ad}{b}+a\lt a+c \lt \dfrac{bc}{d}+c$ or, $\dfrac{\dfrac{ad}{b}+a}{b+d}+\dfrac{a+c}{b+d}\lt \dfrac{\dfrac{bc}{d}+c}{b+d}$ Thus, $\dfrac{a}{b}\lt \dfrac{a+c}{b+d}\lt \dfrac{c}{d}$
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