What is sqrt(3-x)-sqrt(x+1)> 1/2 ?

The solution to sqrt(3-x)-sqrt(x+1)> 1/2 is -1<= x<-(sqrt(31))/8+1

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sqrt(3-x)-sqrt(x+1)> 1/2

3 - x - x + 1 > \frac{1}{2}

- 1 \leq x < - \frac{31}{8} + 1

Solution steps

3 - x - x + 1 > \frac{1}{2}

Add

x + 1

to both sides

3 - x - x + 1 + x + 1 > \frac{1}{2} + x + 1

Simplify

3 - x > \frac{1}{2} + x + 1

f (x) > g (x)

then

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

For

\frac{1}{2} + x + 1 < 0 :

No Solution

For

\frac{1}{2} + x + 1 \geq 0 : x < - \frac{31}{8} + 1

Find singularity points

Find non-negative values for radicals:

- 1 \leq x \leq 3

Combine the intervals

(x < - \frac{31}{8} + 1 or No Solution) and - 1 \leq x \leq 3

Merge Overlapping Intervals

- 1 \leq x < - \frac{31}{8} + 1

Verify Solutions:

- 1 \leq x < - \frac{31}{8} + 1

True

The solution is

- 1 \leq x < - \frac{31}{8} + 1

x + \frac{1}{x} \geq 1

4 x y + 12 y - 4 x - 12

x \cdot e^{- x}

5 x - 5 = 9 x + 3

36 = 54 - 9 a

What is sqrt(3-x)-sqrt(x+1)> 1/2 ?
The solution to sqrt(3-x)-sqrt(x+1)> 1/2 is -1<= x<-(sqrt(31))/8+1