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

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

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

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

x < 1 or \frac{3}{2} < x \leq 2

Interval Notation

(- \infty, 1) \cup (\frac{3}{2}, 2]

Decimal Notation

x < 1 or 1.5 < x \leq 2

Solution steps

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

Rewrite in standard form

\frac{2 x - 4}{( x - 1 ) ( 2 x - 3 )} \leq 0

Factor

\frac{2 x - 4}{( x - 1 ) ( 2 x - 3 )} : \frac{2 ( x - 2 )}{( x - 1 ) ( 2 x - 3 )}

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

Divide both sides by

2

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

Simplify

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

Identify the intervals

x < 1 or \frac{3}{2} < x \leq 2

(x^{4} - 1) (x + 1) \cdot (x - 1)^{2} \geq 0

- x^{3} + x^{2} - 2 x \geq 0

(x + 1) (x + 6) < 0

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

\frac{( - x ^{2} + 4 x - 3 )}{( 2 x - 8 )} > 0

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