What is n-sqrt(3)>= sqrt(3n)+1 ?

The solution to n-sqrt(3)>= sqrt(3n)+1 is n>= 2(2+sqrt(3))

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n-sqrt(3)>= sqrt(3n)+1

n - 3 \geq 3 n + 1

n \geq 2 (2 + 3)

Interval Notation

[2 (2 + 3), \infty)

Decimal Notation

n \geq 7.46410 \dots

Solution steps

n - 3 \geq 3 n + 1

Switch sides

3 n + 1 \leq n - 3

Let

u = 3 n

u + 1 \leq n - 3

u + 1 \leq n - 3 : u \leq n - 3 - 1

u \leq n - 3 - 1

Substitute back

u = 3 n

3 n \leq n - 3 - 1

f (x) \leq g (x)

then

g (x) \geq 0

n - 3 - 1 \geq 0 : n \geq 3 + 1

3 n \leq n - 3 - 1 : n \leq 1 or n \geq 2 (2 + 3)

Combine the intervals

n \geq 3 + 1 and n \geq 2 (2 + 3)

Merge Overlapping Intervals

n \geq 2 (2 + 3)

Find singularity points

Find non-negative values for radicals:

n \geq 0

Combine the intervals

n \geq 2 (2 + 3) and n \geq 0

Merge Overlapping Intervals

n \geq 2 (2 + 3)

x + 6 < x - 6

(m + 2) + (4 m + 5)^{\frac{1}{2}} > 1

x + 3 < x + 33

x < 2

\frac{x}{2} > 4

What is n-sqrt(3)>= sqrt(3n)+1 ?
The solution to n-sqrt(3)>= sqrt(3n)+1 is n>= 2(2+sqrt(3))