What is x+3<sqrt(x+33) ?

The solution to x+3<sqrt(x+33) is -33<= x<3

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x+3<sqrt(x+33)

x + 3 < x + 33

- 33 \leq x < 3

Interval Notation

[- 33, 3)

Solution steps

x + 3 < x + 33

Switch sides

x + 33 > x + 3

f (x) > g (x)

then

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

For

x + 3 < 0 : x < - 3

For

x + 3 \geq 0 : - 3 \leq x < 3

Find singularity points

Find non-negative values for radicals:

x \geq - 33

Combine the intervals

(- 3 \leq x < 3 or x < - 3) and x \geq - 33

Merge Overlapping Intervals

- 33 \leq x < 3

Verify Solutions:

- 33 \leq x < 3

True

The solution is

- 33 \leq x < 3

x < 2

\frac{x}{2} > 4

x < 2 - x

- x^{2} + 8 x - 7 < x - 1

\frac{( ∣ x + 2 ∣ - ∣ x ∣ )}{( 8 - x ^{2} )} \geq 0