First-degree fractional inequality

With a fractional inequality we cannot get rid of the denominator by multiplying: we do not know its sign! A first approach, before introducing the sign table, is to reason by cases about when numerator and denominator have the same sign.

Example — First-degree fractional inequality

Solve $\dfrac{2x+1}{x-3} > 0$.

Warning: we cannot multiply by $x-3$ without knowing its sign! For now we observe that the ratio is positive when numerator and denominator have the same sign:

• Both positive: $2x+1 > 0$ and $x-3 > 0 \implies x > -\frac{1}{2}$ and $x > 3 \implies x > 3$.
• Both negative: $2x+1 < 0$ and $x-3 < 0 \implies x < -\frac{1}{2}$ and $x < 3 \implies x < -\frac{1}{2}$.

Solution: $\boxed{x < -\tfrac{1}{2} \;\text{oppure}\; x > 3}$.

The systematic method (the sign table) will generalise this reasoning to any rational inequality.

Links

Topics: Inequalities
Concepts: Fractional inequality · First-degree inequality · Sign analysis
Skills: Reasoning by cases · Solving inequalities