What is x+5<sqrt(x+7) ?

The solution to x+5<sqrt(x+7) is -7<= x<-3

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x+5<sqrt(x+7)

x + 5 < x + 7

- 7 \leq x < - 3

Interval Notation

[- 7, - 3)

Solution steps

x + 5 < x + 7

Switch sides

x + 7 > x + 5

f (x) > g (x)

then

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

For

x + 5 < 0 : x < - 5

For

x + 5 \geq 0 : - 5 \leq x < - 3

Find singularity points

Find non-negative values for radicals:

x \geq - 7

Combine the intervals

(- 5 \leq x < - 3 or x < - 5) and x \geq - 7

Merge Overlapping Intervals

- 7 \leq x < - 3

Verify Solutions:

- 7 \leq x < - 3

True

The solution is

- 7 \leq x < - 3

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

\frac{( x - 1 )}{( x - 2 )} < 0

\frac{( x ^{2} - 13 x + 40 )}{( 19 x - x ^{2} - 78 )} \leq 0

x - 2 \geq - 1

x - 1 - x < 0