Fractional inequality (2x-1)/(x+3)

Statement

Solve the fractional inequality $\dfrac{2x-1}{x+3} \ge 0$.

Solution

Existence condition: $x \neq -3$.

Study the sign of numerator and denominator:

• Numerator: $2x - 1 \ge 0 \iff x \ge \tfrac12$.
• Denominator: $x + 3 > 0 \iff x > -3$.

The fraction is $\ge 0$ where numerator and denominator have the same sign (or the numerator vanishes):

• for $x < -3$ both negative $\Rightarrow$ fraction positive;
• for $-3 < x < \tfrac12$ opposite signs $\Rightarrow$ fraction negative;
• for $x \ge \tfrac12$ both positive (numerator zero at $\tfrac12$) $\Rightarrow$ fraction $\ge 0$.

$\boxed{x < -3 \ \vee\ x \ge \tfrac12}$