Inequality proof as a part of calculus lesson As part of a calculus lesson I was required to prove that:
(1) if $\ |x-3| < \frac{1}{2},\ $ then $\ \bigg|\displaystyle{\frac{\sin(x^2 -8x+15)}{4x-7}}\bigg| < \frac{1}{2}$
So, by using $|\sin(t)| \le |t|,$ I can prove that:
$$\bigg|\frac{\sin(x^2 -8x+15)}{4x-7}\bigg| \le \frac{5}{6}|x-3|$$
After proving this inequality I assume that the initial requirement is proved, and therefore I'm done, is that it or am I missing something?
Thanks for your assistance :)

HINT.-You want to have $\dfrac52\lt x\lt \dfrac 72$ and your function $|f(x)|=\ \bigg|\displaystyle{\frac{\sin(x^2 -8x+15)}{4x-7}}\bigg| < \frac{1}{2}$ is 
decreasing in this interval from $|f(\frac 52)|=|0.39952128217|$ till
$|f(\frac 72)|=|-0.102734043987|$
