4-(3/x)<x

Solution

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

0 < x < 1 or x > 3

Interval Notation

(0, 1) \cup (3, \infty)

Solution steps

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

Rewrite in standard form

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

Factor

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

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

Multiply both sides by

- 1

(reverse the inequality)

\frac{( - ( x - 1 ) ( x - 3 ) ) ( - 1 )}{x} > 0 \cdot (- 1)

Simplify

\frac{( x - 1 ) ( x - 3 )}{x} > 0

Identify the intervals

0 < x < 1 or x > 3

\frac{( 2 x ^{2} - 1 )}{( 2 x ^{2} + 7 x - 15 )} > 1

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

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

(a + 3) (a - 2) < 0

\frac{2}{( x - 2 )} \geq \frac{1}{x}