|sin(x)|

+ inequation

Solve the inequality!

A inequation with variable

The solution

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|sin(x)| < \/ 3/2 

$$\left|{\sin{\left(x \right)}}\right| < \sqrt{\frac{3}{2}}$$

Abs(sin(x)) < sqrt(3/2)

Detail solution

Given the inequality:
$$\left|{\sin{\left(x \right)}}\right| < \sqrt{\frac{3}{2}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\sin{\left(x \right)}}\right| = \sqrt{\frac{3}{2}}$$
Solve:
Given the equation
$$\left|{\sin{\left(x \right)}}\right| = \sqrt{\frac{3}{2}}$$
transform
$$\left|{\sin{\left(x \right)}}\right| - \frac{\sqrt{6}}{2} - 1 = 0$$
$$\left|{\sin{\left(x \right)}}\right| - \frac{\sqrt{6}}{2} - 1 = 0$$
Do replacement
$$w = \left|{\sin{\left(x \right)}}\right|$$
Expand brackets in the left part

-1 + w - sqrt6/2 = 0

Move free summands (without w)
from left part to right part, we given:
$$w - \frac{\sqrt{6}}{2} = 1$$
Divide both parts of the equation by (w - sqrt(6)/2)/w

w = 1 / ((w - sqrt(6)/2)/w)

We get the answer: w = 1 + sqrt(6)/2
do backward replacement
$$\left|{\sin{\left(x \right)}}\right| = w$$
substitute w:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$\left|{\sin{\left(0 \right)}}\right| < \sqrt{\frac{3}{2}}$$

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    \/ 6 
0 < -----
      2  
    

so the inequality is always executed

Solving inequality on a graph