What is 3/((sqrt(2-x)))-sqrt(2-x)<2 ?

The solution to 3/((sqrt(2-x)))-sqrt(2-x)<2 is x<1

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3/((sqrt(2-x)))-sqrt(2-x)<2

\frac{3}{( 2 - x )} - 2 - x < 2

x < 1

Interval Notation

(- \infty, 1)

Solution steps

\frac{3}{2 - x} - 2 - x < 2

Let

u = 2 - x

\frac{3}{u} - u < 2

\frac{3}{u} - u < 2 : - 3 < u < 0 or u > 1

- 3 < u < 0 or u > 1

Substitute back

u = 2 - x

- 3 < 2 - x < 0 or 2 - x > 1

- 3 < 2 - x < 0 :

No Solution

2 - x > 1 : x < 1

Combine the intervals

No Solution or x < 1

Merge Overlapping Intervals

x < 1

Find singularity points

Find non-negative values for radicals:

x \leq 2

Combine the intervals

x < 1 and x \leq 2

Merge Overlapping Intervals

x < 1

x - 5 - 9 - x > 1

x^{2} < x + 1

x > 1 - x

x + 5 < x + 7

x^{2} - 6 x + 8 \leq x + 1

What is 3/((sqrt(2-x)))-sqrt(2-x)<2 ?
The solution to 3/((sqrt(2-x)))-sqrt(2-x)<2 is x<1