What is (-x^2+4)/x >= 3 ?

The solution to (-x^2+4)/x >= 3 is x<=-4\lor 0<x<= 1

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(-x^2+4)/x >= 3

\frac{- x ^{2} + 4}{x} \geq 3

x \leq - 4 or 0 < x \leq 1

Interval Notation

(- \infty, - 4] \cup (0, 1]

Solution steps

\frac{- x ^{2} + 4}{x} \geq 3

Rewrite in standard form

\frac{- x ^{2} + 4 - 3 x}{x} \geq 0

Factor

\frac{- x ^{2} + 4 - 3 x}{x} : \frac{- ( x - 1 ) ( x + 4 )}{x}

\frac{- ( x - 1 ) ( x + 4 )}{x} \geq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - 1 ) ( x + 4 ) ) ( - 1 )}{x} \leq 0 \cdot (- 1)

Simplify

\frac{( x - 1 ) ( x + 4 )}{x} \leq 0

Identify the intervals

x \leq - 4 or 0 < x \leq 1

2 x + 4 \geq 3 x - 3

x^{2} + x + 5 = 0

\frac{- x + 1}{4} > x + 4

\frac{2}{5} \cdot 40