What is ((x^2-4x+7))/((x^2-7x+12))< 2/3 ?

The solution to ((x^2-4x+7))/((x^2-7x+12))< 2/3 is -3<x<1\lor 3<x<4

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((x^2-4x+7))/((x^2-7x+12))< 2/3

\frac{( x ^{2} - 4 x + 7 )}{( x ^{2} - 7 x + 12 )} < \frac{2}{3}

- 3 < x < 1 or 3 < x < 4

Interval Notation

(- 3, 1) \cup (3, 4)

Solution steps

\frac{x ^{2} - 4 x + 7}{x ^{2} - 7 x + 12} < \frac{2}{3}

Rewrite in standard form

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

Factor

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

\frac{( x - 1 ) ( x + 3 )}{( x - 3 ) ( x - 4 )} < 0

Identify the intervals

- 3 < x < 1 or 3 < x < 4

(x - 2) \cdot (x - 4) > 0

\frac{( 3 x - 1 )}{( x + 2 )} => 5

x^{3} - 6 x < 0

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

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

What is ((x^2-4x+7))/((x^2-7x+12))< 2/3 ?
The solution to ((x^2-4x+7))/((x^2-7x+12))< 2/3 is -3<x<1\lor 3<x<4