What is x/2+6/x <7+x ?

The solution to x/2+6/x -7+sqrt(61)

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x/2+6/x <7+x

\frac{x}{2} + \frac{6}{x} < 7 + x

- 7 - 61 < x < 0 or x > - 7 + 61

Interval Notation

(- 7 - 61, 0) \cup (- 7 + 61, \infty)

Decimal Notation

- 14.81024 \dots < x < 0 or x > 0.81024 \dots

Solution steps

\frac{x}{2} + \frac{6}{x} < 7 + x

Rewrite in standard form

\frac{- x ^{2} - 14 x + 12}{x} < 0

Factor

- x^{2} - 14 x + 12 : - (x + 7 - 61) (x + 7 + 61)

\frac{- ( x + 7 - 61 ) ( x + 7 + 61 )}{x} < 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x + 7 - 61 ) ( x + 7 + 61 ) ) ( - 1 )}{x} > 0 \cdot (- 1)

Simplify

\frac{( x + 7 - 61 ) ( x + 7 + 61 )}{x} > 0

Identify the intervals

- 7 - 61 < x < 0 or x > - 7 + 61

\frac{x - 2}{x - 3} < 0

x^{2} (x - 3) (x - 1) > 0

3 x^{2} + 5 x - 2 - x^{2} + 10 x - 25 > 0

(a - 1)^{2} (a + 1) > 0

x (x + 6) \cdot (x + 2)^{2} \cdot (x - 3) > 0

What is x/2+6/x <7+x ?
The solution to x/2+6/x <7+x is -7-sqrt(61)<x<0\lor x>-7+sqrt(61)