Question

Solve the following inequality:
${\log }_{(x-4.5)}\frac{x+4}{2x-6}⩽{\log }_{(x-4.5)}(x-5)$

Answer 1

For $\log \frac{x+4}{2x-6}\Rightarrow$ $\frac{x+4}{2x-6}>0(x+4)(2x-6)>0x\in (-\infty ,-4)\cup (3,\infty )$

For $\log (x-5)\Rightarrow$ $x-5>0x>5$

For $\log (x-4.5)\Rightarrow$ $x-4.5>0x>4.5$

Taking intersection, we get $x\in (5,\infty )$

For $x>5.5$

$\log \frac{x+4}{2x-6}\le \log (x-5)\frac{x+4}{2x-6}\le x-5x+4\le (x-5)(2x-6)x+4\le 2x^2-16x+302x^2-17x+26\ge 0x\in (-\infty ,-2]\cup [6.5,\infty )Hence,x\in [6.5,\infty )$

For $x\in (4.5,5.5)$

$\log \frac{x+4}{2x-6}\ge \log (x-5)\frac{x+4}{2x-6}\ge x-5x+4\ge (x-5)(2x-6)x+4\ge 2x^2-16x+302x^2-17x+26\le 0x\in [2,6.5]Hence,x\in (4.5,5.5)$

Taking union, we get $x\in (5,5.5)\cup [6.5,\infty )$