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

The solution to (x-3)/x >= 4 is -1<= x<0

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(x-3)/x >= 4

\frac{x - 3}{x} \geq 4

- 1 \leq x < 0

Interval Notation

[- 1, 0)

Solution steps

\frac{x - 3}{x} \geq 4

Rewrite in standard form

\frac{- 3 x - 3}{x} \geq 0

Factor

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

\frac{- 3 ( x + 1 )}{x} \geq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - 3 ( x + 1 ) ) ( - 1 )}{x} \leq 0 \cdot (- 1)

Simplify

\frac{3 ( x + 1 )}{x} \leq 0

Divide both sides by

3

\frac{\frac{3 ( x + 1 )}{x}}{3} \leq \frac{0}{3}

Simplify

\frac{x + 1}{x} \leq 0

Identify the intervals

- 1 \leq x < 0

x^{2} - 4 x + 6 \geq 0

\frac{( n ^{2} + 2 )}{( n ^{2} - 1 )} + 2 < 0

3 - x^{2} < 0

\frac{( x ^{2} \cdot ( x - 2 ) ( 5 - x ) )}{( ( x ^{2} - 4 ) ( x - 1 ) )} > 0

\frac{( 6 x - x ^{2} )}{( ( x + 9 ) ^{2} )} \leq 0