Given the equation
$$4 \left|{x}\right| + \frac{1}{\left|{x}\right|} = 4 \sin{\left(\frac{\pi x}{4} \right)}$$
transform
$$- 4 \sin{\left(\frac{\pi x}{4} \right)} + 4 \left|{x}\right| + \frac{1}{\left|{x}\right|} = 0$$
$$\left(4 \left|{x}\right| + \frac{1}{\left|{x}\right|}\right) - 4 \sin{\left(\frac{\pi x}{4} \right)} = 0$$
Do replacement
$$w = \sin{\left(\frac{\pi x}{4} \right)}$$
Given the equation:
$$\left(4 \left|{x}\right| + \frac{1}{\left|{x}\right|}\right) - 4 \sin{\left(\frac{\pi x}{4} \right)} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
a1 = 1
b1 = |x|
a2 = 1
b2 = 1/(-4*|x| + 4*sin(pi*x/4))
so we get the equation
$$\frac{1}{4 \sin{\left(\frac{\pi x}{4} \right)} - 4 \left|{x}\right|} = \left|{x}\right|$$
$$\frac{1}{4 \sin{\left(\frac{\pi x}{4} \right)} - 4 \left|{x}\right|} = \left|{x}\right|$$
Expand brackets in the left part
-1/4*+1/x+1/4*sin+1/pi*x/4) = |x|
This equation has no roots
do backward replacement
$$\sin{\left(\frac{\pi x}{4} \right)} = w$$
Given the equation
$$\sin{\left(\frac{\pi x}{4} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{\pi x}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$\frac{\pi x}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$\frac{\pi x}{4} = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$\frac{\pi x}{4} = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{\pi}{4}$$
substitute w: