What is (4x)/(x-1)<= x/(x+6)+4 ?

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(4x)/(x-1)<= x/(x+6)+4

\frac{4 x}{x - 1} \leq \frac{x}{x + 6} + 4

x < - 6 or - 3 \leq x < 1 or x \geq 8

Interval Notation

(- \infty, - 6) \cup [- 3, 1) \cup [8, \infty)

Solution steps

\frac{4 x}{x - 1} \leq \frac{x}{x + 6} + 4

Rewrite in standard form

\frac{- x ^{2} + 5 x + 24}{( x - 1 ) ( x + 6 )} \leq 0

Factor

\frac{- x ^{2} + 5 x + 24}{( x - 1 ) ( x + 6 )} : \frac{- ( x + 3 ) ( x - 8 )}{( x - 1 ) ( x + 6 )}

\frac{- ( x + 3 ) ( x - 8 )}{( x - 1 ) ( x + 6 )} \leq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x + 3 ) ( x - 8 ) ) ( - 1 )}{( x - 1 ) ( x + 6 )} \geq 0 \cdot (- 1)

Simplify

\frac{( x + 3 ) ( x - 8 )}{( x - 1 ) ( x + 6 )} \geq 0

Identify the intervals

x < - 6 or - 3 \leq x < 1 or x \geq 8

(3 w - 5) w = 50

(2)^{2}

∣ x + 3 ∣ \leq - 8

(x - 5)^{2} = 3

\frac{5}{12} + \frac{3}{4}

What is (4x)/(x-1)<= x/(x+6)+4 ?
The solution to (4x)/(x-1)<= x/(x+6)+4 is x<-6\lor-3<= x<1\lor x>= 8