Question

For the inequation
$(1.25{)}^{1-x}<(0.64{)}^{2(1+\sqrt{x})}$

• A. There is no real value of $x$ exists
• B. There are infinitely man negative values of $x$ exist
• C. Any real value of $x$ greater than $25$ satisfies the inequation
• D. There is exactly one positive and one negative integer only satisfying the equation

Answer 1

The correct option is C Any real value of $x$ greater than $25$ satisfies the inequation

${\left(\frac{5}{4}\right)}^{1-x}<{\left(\frac{16}{25}\right)}^{2(1+\sqrt{x})}$
${\left(\frac{4}{5}\right)}^{x-1}<{\left(\frac{4}{5}\right)}^{4(1+\sqrt{x})}$
Now $0<\frac{4}{5}<1\Rightarrow x-1>4(1+\sqrt{x})$
$x-4\sqrt{x}-5>0\Rightarrow (\sqrt{x}-\sqrt{5})(\sqrt{x}+1)>0$
$\Rightarrow \sqrt{x}>5\Rightarrow x>25$