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x^2-1
Solving inequalities/ x^2-1

x^2-1 inequation

A inequation with variable

The solution

You have entered [src] 
 2        
x  - 1 > 0
x2−1>0x^{2} - 1 > 0 
x^2 - 1*1 > 0
Detail solution
Given the inequality:
x2−1>0x^{2} - 1 > 0
To solve this inequality, we must first solve the corresponding equation:
x2−1=0x^{2} - 1 = 0
Solve:
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=1a = 1
b=0b = 0
c=−1c = -1
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-1) = 4

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

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

or
x1=1x_{1} = 1
Simplify
x2=−1x_{2} = -1
Simplify
x1=1x_{1} = 1
x2=−1x_{2} = -1
x1=1x_{1} = 1
x2=−1x_{2} = -1
This roots
x2=−1x_{2} = -1
x1=1x_{1} = 1
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}
=
−1−110-1 - \frac{1}{10}
=
−1110- \frac{11}{10}
substitute to the expression
x2−1>0x^{2} - 1 > 0
(−1)1+(−1110)2>0\left(-1\right) 1 + \left(- \frac{11}{10}\right)^{2} > 0
 21    
--- > 0
100

one of the solutions of our inequality is:
x<−1x < -1
 _____           _____          
      \         /
-------ο-------ο-------
       x_2      x_1

Other solutions will get with the changeover to the next point
etc.
The answer:
x<−1x < -1
x>1x > 1
Solving inequality on a graph
-5.0-4.0-3.0-2.0-1.05.00.01.02.03.04.0-2020
Rapid solution [src] 
Or(And(-oo < x, x < -1), And(1 < x, x < oo))
(−∞<x∧x<−1)∨(1<x∧x<∞)\left(-\infty < x \wedge x < -1\right) \vee \left(1 < x \wedge x < \infty\right) 
((-oo < x)∧(x < -1))∨((1 < x)∧(x < oo))
Rapid solution 2 [src] 
(-oo, -1) U (1, oo)
x in (−∞,−1)∪(1,∞)x\ in\ \left(-\infty, -1\right) \cup \left(1, \infty\right) 
x in Union(Interval.open(-oo, -1), Interval.open(1, oo))
The graph
x^2-1 inequation
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