The solution to 1/x = (1+sqrt(5))/2

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1/x <= x-1

Solution

\frac{1}{x} \leq x - 1

\frac{1 - 5}{2} \leq x < 0 or x \geq \frac{1 + 5}{2}

Interval Notation

[\frac{1 - 5}{2}, 0) \cup [\frac{1 + 5}{2}, \infty)

Decimal Notation

- 0.61803 \dots \leq x < 0 or x \geq 1.61803 \dots

Solution steps

\frac{1}{x} \leq x - 1

Rewrite in standard form

\frac{1 + x - x ^{2}}{x} \leq 0

Factor

1 + x - x^{2} : - (x - \frac{1 + 5}{2}) (x - \frac{1 - 5}{2})

\frac{- ( x - \frac{1 + 5}{2} ) ( x - \frac{1 - 5}{2} )}{x} \leq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - \frac{1 + 5}{2} ) ( x - \frac{1 - 5}{2} ) ) ( - 1 )}{x} \geq 0 \cdot (- 1)

Simplify

\frac{( x - \frac{1 + 5}{2} ) ( x - \frac{1 - 5}{2} )}{x} \geq 0

Identify the intervals

\frac{1 - 5}{2} \leq x < 0 or x \geq \frac{1 + 5}{2}

3 5 - \frac{7}{3} x + 7 = 13

8 x - 2 < 8

3 x^{3} - 5 x^{2} - 12 x + 20

- 14 > w + 3

What is 1/x <= x-1 ?
The solution to 1/x <= x-1 is (1-sqrt(5))/2 <= x<0\lor x>= (1+sqrt(5))/2