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Solving inequalities/ 16*x^2<47

16*x^2<47 inequation

A inequation with variable

The solution

You have entered [src] 
    2     
16*x  < 47
16x2<4716 x^{2} < 47 
16*x^2 < 47
Detail solution
Given the inequality:
16x2<4716 x^{2} < 47
To solve this inequality, we must first solve the corresponding equation:
16x2=4716 x^{2} = 47
Solve:
Move right part of the equation to
left part with negative sign.

The equation is transformed from
16x2=4716 x^{2} = 47
to
16x2−47=016 x^{2} - 47 = 0
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=16a = 16
b=0b = 0
c=−47c = -47
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (16) * (-47) = 3008

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=474x_{1} = \frac{\sqrt{47}}{4}
x2=−474x_{2} = - \frac{\sqrt{47}}{4}
x1=474x_{1} = \frac{\sqrt{47}}{4}
x2=−474x_{2} = - \frac{\sqrt{47}}{4}
x1=474x_{1} = \frac{\sqrt{47}}{4}
x2=−474x_{2} = - \frac{\sqrt{47}}{4}
This roots
x2=−474x_{2} = - \frac{\sqrt{47}}{4}
x1=474x_{1} = \frac{\sqrt{47}}{4}
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
x0<x2x_{0} < x_{2}
For example, let's take the point
x0=x2−110x_{0} = x_{2} - \frac{1}{10}
=
−474−110- \frac{\sqrt{47}}{4} - \frac{1}{10}
=
−474−110- \frac{\sqrt{47}}{4} - \frac{1}{10}
substitute to the expression
16x2<4716 x^{2} < 47
16(−474−110)2<4716 \left(- \frac{\sqrt{47}}{4} - \frac{1}{10}\right)^{2} < 47
                  2     
   /         ____\      
   |  1    \/ 47 |  < 47
16*|- -- - ------|      
   \  10     4   /

but
                  2     
   /         ____\      
   |  1    \/ 47 |  > 47
16*|- -- - ------|      
   \  10     4   /

Then
x<−474x < - \frac{\sqrt{47}}{4}
no execute
one of the solutions of our inequality is:
x>−474∧x<474x > - \frac{\sqrt{47}}{4} \wedge x < \frac{\sqrt{47}}{4}
         _____  
        /     \  
-------ο-------ο-------
       x2      x1
Solving inequality on a graph
01234567-7-6-5-4-3-2-101000
Rapid solution [src] 
   /   ____             ____\
   |-\/ 47            \/ 47 |
And|-------- < x, x < ------|
   \   4                4   /
−474<x∧x<474- \frac{\sqrt{47}}{4} < x \wedge x < \frac{\sqrt{47}}{4} 
(-sqrt(47)/4 < x)∧(x < sqrt(47)/4)
Rapid solution 2 [src] 
    ____     ____ 
 -\/ 47    \/ 47  
(--------, ------)
    4        4
x in (−474,474)x\ in\ \left(- \frac{\sqrt{47}}{4}, \frac{\sqrt{47}}{4}\right) 
x in Interval.open(-sqrt(47)/4, sqrt(47)/4)
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