What is ((x-1))/((x-2))+((x+1))/((3-x))<= 1 ?

The solution to ((x-1))/((x-2))+((x+1))/((3-x)) 3

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((x-1))/((x-2))+((x+1))/((3-x))<= 1

\frac{( x - 1 )}{( x - 2 )} + \frac{( x + 1 )}{( 3 - x )} \leq 1

x < 2 or x > 3

Interval Notation

(- \infty, 2) \cup (3, \infty)

Solution steps

\frac{x - 1}{x - 2} + \frac{x + 1}{3 - x} \leq 1

Rewrite in standard form

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

Factor

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

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

Identify the intervals

x < 2 or x > 3

9 x^{2} - 20 x + 11 > 0

x^{2} \cdot (x - 2)^{4} \cdot (x - 3) \geq 0

x^{2} + x + 2 < 0

\frac{x}{2} > \frac{5}{x + 1 + 4}

\frac{( x + 1 )}{( x + 2 )} \geq 1

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