Number of negative integral values of x satisfying $4^{-x+0.5}-{7.2}^{-x}<4,x\in R\text{ is }$

$4^{-x+0.5}-{7.2}^{-x}<4------\left(1\right)$

$\text{ Let }{2}^{-x}=t\therefore t>0----\left(2\right)$

$\begin{array}{l}\therefore \text{ (1) reduces to }2t^2-7t<4\\ \Rightarrow 2t^2-7t-4<0\\ \Rightarrow (2t+1)(t-4)<0\end{array}$

$\Rightarrow \frac{-1}{2}<t<4---\left(3\right)$

$\begin{array}{r}\text{⇒ from (2) and (3) }\\ 0<t<4 \\ \Rightarrow 0<2^{-x}<4 \left(\sin ce 2^{-x}<4 \Rightarrow -x< \log {}_{2}4\Rightarrow -x<2 \Rightarrow x>-2\right) \\ \Rightarrow -2<x<\infty \\ \Rightarrow x\in (-2,\mathrm{\infty }) \Rightarrow x=-1 is the only negative value \\ \\ \end{array}$