What is sqrt(x+1)+sqrt(x-1)<sqrt(3x) ?

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

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sqrt(x+1)+sqrt(x-1)<sqrt(3x)

x + 1 + x - 1 < 3 x

1 \leq x < \frac{2}{3}

Interval Notation

[1, \frac{2}{3})

Decimal Notation

1 \leq x < 1.15470 \dots

Solution steps

x + 1 + x - 1 < 3 x

Apply radical rule:

assuming

a \geq 0, b \geq 0

x + 1 + x - 1 < 3 x

Move

x - 1

to the right side

x + 1 < 3 x - x - 1

f (x) < g (x)

then

g (x) > 0

3 x - x - 1 > 0 : x > - \frac{1}{2}

x + 1 < 3 x - x - 1 : \frac{2}{3} < x < \frac{2}{3}

Find singularity points

Find non-negative values for radicals:

x \geq 1

Combine the intervals

x > - \frac{1}{2} and \frac{2}{3} < x < \frac{2}{3} and x \geq 1

Merge Overlapping Intervals

1 \leq x < \frac{2}{3}

Verify Solutions:

1 \leq x < \frac{2}{3}

True

The solution is

1 \leq x < \frac{2}{3}

30 v + 50 x

x (3 x - 18) (6 x - 54) \geq 0

17 - v < 13

\frac{3}{4} x = \frac{2}{3}

12 g^{2} - 37 g + 28

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