1< 3/x — step-by-step solution

1) Let's solve in

ℝ

1 < \frac{3}{x}

\frac{3}{x} > 1

2) Let's sum the fractions:

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

ⓘ To add fractions, put them over the same denominator, then add the numerators:

\frac{3}{x} - 1

\frac{3 - 1 \cdot x}{x}

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

ⓘ Multiplying by 1 changes nothing:

1 \cdot x

x

\frac{3 - x}{x} > 0

We study the sign of the numerator

⇥The expression

3 - x

is zero at:

⇥

x = 3

⇥Its coefficient of

x

-1

, being negative, it is positive before and negative after

3

We study the sign of the denominator, which must not be zero

⇥The expression

x

is zero at

x = 0

(forbidden value), it is negative before and positive after

3) Let's draw up the sign table:

Candidates, pending the domain check:

x \in (0, 3)

For the expression

\frac{3}{x}

to exist, you must check:

x \neq 0

Domain constraints:

x \neq 0

▶Solution:

x \in (0, 3)