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Identical expressions
- four *(|x|)+ one /(|x|)= four *sin(pi*x/ four)
- 4 multiply by ( module of x|) plus 1 divide by (|x|) equally 4 multiply by sinus of ( Pi multiply by x divide by 4)
- four multiply by ( module of x|) plus one divide by (|x|) equally four multiply by sinus of ( Pi multiply by x divide by four)
- 4(|x|)+1/(|x|)=4sin(pix/4)
- 4|x|+1/|x|=4sinpix/4
- 4*(|x|)+1 divide by (|x|)=4*sin(pi*x divide by 4)
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Similar expressions
- 4*(|x|)-1/(|x|)=4*sin(pi*x/4)
4*(|x|)+1/(|x|)=4*sin(pi*x/4) equation
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The solution
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1 /pi*x\
4*|x| + --- = 4*sin|----|
|x| \ 4 /
$$4 \left|{x}\right| + \frac{1}{\left|{x}\right|} = 4 \sin{\left(\frac{\pi x}{4} \right)}$$
Detail solution
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
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
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:
$$4 \left|{x}\right| + \frac{1}{\left|{x}\right|} = 4 \sin{\left(\frac{\pi x}{4} \right)}$$
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: