Question

Solve the following inequalities.
${25}^{-x}-5^{-x+1}⩾50$

Answer 1

${25}^{-x}-5^{-x+1}\ge 50$

$5^{-2x}-5^{-x}\ge 50$ $[\because a^{m+m}=a^m.a^n]$

Let $5^x=a$

$\Rightarrow a^2-5a-50\ge 0$

$\Rightarrow a^2+5a-10a-50\ge 0$

$\Rightarrow a(a+5)-10(a+5)\ge 0$

$\Rightarrow (a+5)(a-10)\ge 0$

$\Rightarrow a\le -5$ (or) $a\ge 10$ $[\because (x-a\left)\right(x-b)>0a<b]$

$\Rightarrow x<a$ or $x>b$

$\Rightarrow 5^{-x}\le -5$ (or) $5^{-x}\ge 10$

$\Rightarrow -xlog\le -log5$

$x\le 1$

$-xlog5\ge log10$

$-xlog5\ge 1$

$-x\ge -log5$

$x\ge log5$