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

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

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

x - 1 < \frac{1}{( x - 3 )}

x < 2 - 2 or 3 < x < 2 + 2

Interval Notation

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

Decimal Notation

x < 0.58578 \dots or 3 < x < 3.41421 \dots

Solution steps

x - 1 < \frac{1}{x - 3}

Rewrite in standard form

\frac{x ^{2} - 4 x + 2}{x - 3} < 0

Factor

x^{2} - 4 x + 2 : (x - 2 - 2) (x - 2 + 2)

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

Identify the intervals

x < 2 - 2 or 3 < x < 2 + 2

\frac{( x + 2 ) ( x ^{2} - 2 x + 1 )}{( - 4 + 3 x - x ^{2} )} \geq 0

x^{2} > - 4

x^{2} > - 2

48 + 2 x - x^{2} \geq 0

\frac{5}{6 ( x + 2 ) - 4} < \frac{x}{3 + 12}

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