What is x-1<sqrt(x^2+1) ?

The solution to x-1<sqrt(x^2+1) is True for all x

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x-1<sqrt(x^2+1)

x - 1 < x^{2} + 1

True for all x

Interval Notation

(- \infty, \infty)

Solution steps

x - 1 < x^{2} + 1

Switch sides

x^{2} + 1 > x - 1

f (x) > g (x)

then

g (x) < 0 or g (x) \geq 0

For

x - 1 < 0 : x < 1

For

x - 1 \geq 0 : x \geq 1

Combine the intervals

x \geq 1 or x < 1

Merge Overlapping Intervals

True for all x

Verify Solutions:

The solution is

True for all x

2 + x - x^{2} < x + 2

2 x + 14 > x + 3

\frac{( x - 3 )}{( x - 2 )} > 1

x > (2 - x)^{\frac{1}{2}}

x - 1 - 8 - 2 x \geq 1