by HallsofIvy » Sat Mar 09, 2019 3:34 pm
Equivalently, assume for the moment that [tex]\frac{15}{8}> \frac{17}{6}[/tex]. Since the denominators are both positive, we can multiply both sides by 8 and 6 without changing the direction of the inequality: (15)(6)> (17)(8) or 90> 136. That is NOT true so our original assumption, that [tex]\frac{15}{8}> \frac{17}{6}[/tex], is NOT true. Those are also not equal so we must have [tex]\frac{15}{8}< \frac{17}{6}[/tex].
A formal proof of that inequality would reverse that calculation. From the obviously true "90< 136", divide both sides by the (positive) numbers 8 and 6 to get [tex]\frac{90}{(8)(6)}< \frac{136}{(8)(6)}[/tex] or [tex]\frac{15}{8}< \frac{17}{6}[/tex].