How do I solve this inequality? $3\sin(2x)> \sin(x)+\cos(x)+1$

Question

How do I solve this inequality?

$3\sin(2x)> \sin(x)+\cos(x)+1$

I am unable to think of a method to solve this.

Writing $\sin(2x)=2\sin(x)\cos(x)$ don't seem to help

Answer 1

Hint

$$\sin x+\cos x=u,u^2=1+\sin2x$$

$$3(u^2-1)>u+1$$

$$\iff3u^2-u-4>0$$

$$\iff(3u-4)(u+1)>0$$

$\implies$ either $u>$max$\left(-1,\dfrac43\right)=?$

or $u<$min$\left(-1,\dfrac43\right)=?$

Now keep in mind $\sin x+\cos x=\sqrt2\sin\left(x+\dfrac\pi4\right)\implies-\sqrt2\le u\le\sqrt2$