Question

The values of $\frac{1}{x}$ for x ∈ (-2, 4) is _________.

• A.
• B.
• C. U (,)
• D. U

Answer 1

The correct option is C

U (,)

Given that X ∈ (-2,4) - {0} [as $\frac{1}{x}$ is not defined for x = 0]

This also can be written as X ∈ (-2,0) U (0,4)

For X ∈ (-2,0) $\Rightarrow$ -2 < x < 0

$\Rightarrow$ $\frac{1}{-2}$ > $\frac{1}{x}$ > $\frac{1}{0^{-}}$

$0^{-}$ means that x approaches 0 from -ve side On the number line.

$\Rightarrow$ $-\frac{1}{2}$ > $\frac{1}{x}$ > $-\infty$

$\Rightarrow$ $-\infty$ < $\frac{1}{x}$ < $\frac{-1}{2}$ ----------------(1)

For X ∈ (0,4) $\Rightarrow$ 0 < x < 4 [$0^{+}$ means that x approaches 0 from +ve side on the number line]

$\Rightarrow$ $\frac{1}{0^{+}}$ > $\frac{1}{x}$ > $\frac{1}{4}$

$\Rightarrow$ $+\infty$ > $\frac{1}{x}$ > $\frac{1}{4}$

$\Rightarrow$ $\frac{1}{4}$ < $\frac{1}{x}$ < $+\infty$ ------------(2)

From (1) and (2), $\frac{1}{x}$ ∈ $(-\infty ,\frac{-1}{2})$ U ($\frac{1}{4}$,$\infty$).