What is ((x-5))/(3-x)>=-3(x-1) ?

The solution to ((x-5))/(3-x)>=-3(x-1) is 2 3

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((x-5))/(3-x)>=-3(x-1)

\frac{( x - 5 )}{3 - x} \geq - 3 (x - 1)

2 \leq x \leq \frac{7}{3} or x > 3

Interval Notation

[2, \frac{7}{3}] \cup (3, \infty)

Decimal Notation

2 \leq x \leq 2.33333 \dots or x > 3

Solution steps

\frac{x - 5}{3 - x} \geq - 3 (x - 1)

Rewrite in standard form

\frac{- 3 x ^{2} + 13 x - 14}{3 - x} \geq 0

Factor

\frac{- 3 x ^{2} + 13 x - 14}{3 - x} : \frac{- ( x - 2 ) ( 3 x - 7 )}{3 - x}

\frac{- ( x - 2 ) ( 3 x - 7 )}{3 - x} \geq 0

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - 2 ) ( 3 x - 7 ) ) ( - 1 )}{3 - x} \leq 0 \cdot (- 1)

Simplify

\frac{( x - 2 ) ( 3 x - 7 )}{3 - x} \leq 0

Identify the intervals

2 \leq x \leq \frac{7}{3} or x > 3

x^{2} + 3 > 4 x

\frac{( x ^{2} + 2 x - 3 )}{( x ^{2} + 1 )} < 0

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

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

\frac{( x ^{2} - 5 x + 12 )}{( x ^{2} - 4 x + 5 )} > 3

What is ((x-5))/(3-x)>=-3(x-1) ?
The solution to ((x-5))/(3-x)>=-3(x-1) is 2<= x<= 7/3 \lor x>3