What is ((x-4))/((x-1))> x/3+4 ?

The solution to ((x-4))/((x-1))> x/3+4 is x<-8\lor 0<x<1

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((x-4))/((x-1))> x/3+4

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

x < - 8 or 0 < x < 1

Solution steps

\frac{x - 4}{x - 1} > \frac{x}{3} + 4

Rewrite in standard form

\frac{- x ^{2} - 8 x}{x - 1} > 0

Factor

\frac{- x ^{2} - 8 x}{x - 1} : \frac{- x ( x + 8 )}{x - 1}

\frac{- x ( x + 8 )}{x - 1} > 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - x ( x + 8 ) ) ( - 1 )}{x - 1} < 0 \cdot (- 1)

Simplify

\frac{x ( x + 8 )}{x - 1} < 0

Identify the intervals

x < - 8 or 0 < x < 1

(x - 4) (x + 6) < 0

2 (- \frac{4}{3})^{2} + 5 (- \frac{4}{3}) + 6

3 3^{2} + 5 6^{2} = c^{2}

9 + 4 ∣ 3 - 5 x ∣ = 21

2 - x^{2} = 0

What is ((x-4))/((x-1))> x/3+4 ?
The solution to ((x-4))/((x-1))> x/3+4 is x<-8\lor 0<x<1