What is 4/((x-2)^2)>1 ?

The solution to 4/((x-2)^2)>1 is 0<x<2\lor 2<x<4

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4/((x-2)^2)>1

\frac{4}{( x - 2 ) ^{2}} > 1

0 < x < 2 or 2 < x < 4

Interval Notation

(0, 2) \cup (2, 4)

Solution steps

\frac{4}{( x - 2 ) ^{2}} > 1

Rewrite in standard form

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

Factor

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

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

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - x ( x - 4 ) ) ( - 1 )}{( x - 2 ) ^{2}} < 0 \cdot (- 1)

Simplify

\frac{x ( x - 4 )}{( x - 2 ) ^{2}} < 0

Identify the intervals

0 < x < 2 or 2 < x < 4

5 - [2 x + (x + 2)] < 4

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

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

x^{3} - 3 x^{2} - 10 x + 24 > 0

x^{2} + 6 x - 7 \leq 2