What is x-sqrt(1-x)<0 ?

The solution to x-sqrt(1-x)<0 is x<(sqrt(5)-1)/2

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x-sqrt(1-x)<0

x - 1 - x < 0

x < \frac{5 - 1}{2}

Interval Notation

(- \infty, \frac{5 - 1}{2})

Decimal Notation

x < 0.61803 \dots

Solution steps

x - 1 - x < 0

Subtract

x

from both sides

x - 1 - x - x < 0 - x

Simplify

- 1 - x < - x

Multiply both sides by

- 1

(reverse the inequality)

(- 1 - x) (- 1) > (- x) (- 1)

Simplify

1 - x > x

f (x) > g (x)

then

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

For

x < 0 : x < 0

For

x \geq 0 : 0 \leq x < \frac{5 - 1}{2}

Find singularity points

Find non-negative values for radicals:

x \leq 1

Combine the intervals

(0 \leq x < \frac{5 - 1}{2} or x < 0) and x \leq 1

Merge Overlapping Intervals

x < \frac{5 - 1}{2}

Verify Solutions:

x < \frac{5 - 1}{2}

True

The solution is

x < \frac{5 - 1}{2}

x^{2} + x - 12 < x

5 - 2 x < 6 x - 1

6 x - x^{2} < 3 + x

\frac{- 3}{( x ^{2} + 2 x + 10 )} \geq 0

\frac{( x + 5 )}{∣ 1 - x ∣} > 1