Question

Let the domains of $x$ of the given equation ${\log }_a\left(\frac{2x+3}{x}\right)\ge 0$ for (i) $a>1$ be $[-n,\frac{-p}{k})\cup (\frac{p}{k},\infty )$ (ii) $0<a<1$ be $[-\infty ,-m)$. Find $m+n+p+k$

Answer 1

${\log }_a\left(\frac{2x+3}{x}\right)\ge 0\Rightarrow x\ne \frac{-3}{2}$

Since, $a>1$

$\frac{2x+3}{x}\ge 1$

$2x+3\ge x$

$x+3\ge 0$

$x\in [-3,\frac{-3}{2})\cup (\frac{3}{2},\infty ]$

Also,

For $0<a<1$ then

$0<\frac{2x+3}{x}<1$

$2x+3<x$

$x+3<0$

$x\in (-\infty ,-3]$

Given, $m=3,n=3,p=3,k=2$

$m+n+p+k=3+3+3+2=11$