The least positive integral value of x for which $$1x$$−$$2+1x$$−$$1$$>$$1x$$ is true is

$$1x$$−$$2+1x$$−$$1$$>$$1x$$⇒ $$2x2$$−$$3x$$−$$x2+3x$$−$$2(x$$−$$2)(x$$−$$1)x$$>$$0$$⇒ $$x2$$−$$2(x$$−$$2)(x$$−$$1)x$$>$$0$$ x∈$$($$−$$2$$,$$0)$$∪(1$$,$$2)∪(2$$,∞$$)

Introduction to linear inequalities

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Question

The least positive integral value of x for which $\frac{1}{x - 2} + \frac{1}{x - 1} > \frac{1}{x} is true is$

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Solution

$\begin{array}{l} \frac{1}{x - 2} + \frac{1}{x - 1} > \frac{1}{x} \\ \Rightarrow \frac{2 x^{2} - 3 x - x^{2} + 3 x - 2}{(x - 2) (x - 1) x} > 0 \\ \Rightarrow \frac{x^{2} - 2}{(x - 2) (x - 1) x} > 0 \end{array}$

$x \in (- \sqrt{2}, 0) \cup (1, \sqrt{2}) \cup (2, \infty)$

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