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\frac{x^{2}}{x-1}-x\leq 1
Subtract x from both sides.
\frac{x^{2}}{x-1}-\frac{x\left(x-1\right)}{x-1}\leq 1
To add or subtract expressions, expand them to make their denominators the same. Multiply x times \frac{x-1}{x-1}.
\frac{x^{2}-x\left(x-1\right)}{x-1}\leq 1
Since \frac{x^{2}}{x-1} and \frac{x\left(x-1\right)}{x-1} have the same denominator, subtract them by subtracting their numerators.
\frac{x^{2}-x^{2}+x}{x-1}\leq 1
Do the multiplications in x^{2}-x\left(x-1\right).
\frac{x}{x-1}\leq 1
Combine like terms in x^{2}-x^{2}+x.
x-1>0 x-1<0
Denominator x-1 cannot be zero since division by zero is not defined. There are two cases.
x>1
Consider the case when x-1 is positive. Move -1 to the right hand side.
x\leq x-1
The initial inequality does not change the direction when multiplied by x-1 for x-1>0.
x-x\leq -1
Move the terms containing x to the left hand side and all other terms to the right hand side.
0\leq -1
Combine like terms.
x\in \emptyset
Consider condition x>1 specified above.
x<1
Now consider the case when x-1 is negative. Move -1 to the right hand side.
x\geq x-1
The initial inequality changes the direction when multiplied by x-1 for x-1<0.
x-x\geq -1
Move the terms containing x to the left hand side and all other terms to the right hand side.
0\geq -1
Combine like terms.
x<1
Consider condition x<1 specified above.
x<1
The final solution is the union of the obtained solutions.