What is (x-12)/x <= (20)/(1-x) ?

The solution to (x-12)/x <= (20)/(1-x) is -4<= x<=-3\lor 0<x<1

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(x-12)/x <= (20)/(1-x)

\frac{x - 12}{x} \leq \frac{20}{1 - x}

- 4 \leq x \leq - 3 or 0 < x < 1

Solution steps

\frac{x - 12}{x} \leq \frac{20}{1 - x}

Rewrite in standard form

\frac{- x ^{2} - 7 x - 12}{x ( - x + 1 )} \leq 0

Factor

\frac{- x ^{2} - 7 x - 12}{x ( - x + 1 )} : \frac{- ( x + 3 ) ( x + 4 )}{x ( - x + 1 )}

\frac{- ( x + 3 ) ( x + 4 )}{x ( - x + 1 )} \leq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x + 3 ) ( x + 4 ) ) ( - 1 )}{x ( - x + 1 )} \geq 0 \cdot (- 1)

Simplify

\frac{( x + 3 ) ( x + 4 )}{x ( - x + 1 )} \geq 0

Identify the intervals

- 4 \leq x \leq - 3 or 0 < x < 1

2 x^{2} + 7 x = - 3

3 x^{2} - 12 x + 6 = 0

∣ x + 2 ∣ + ∣ x ∣ \leq 3

- 3 x - 5 < 10

6 x + 6 = 24

What is (x-12)/x <= (20)/(1-x) ?
The solution to (x-12)/x <= (20)/(1-x) is -4<= x<=-3\lor 0<x<1