Linear equations and inequalities

Solving first-degree equations and inequalities of the form $ax+b\ \square\ 0$. The equation $ax+b=0$ has the solution:

$x=-\frac{b}{a}\quad(a\neq0)$

For an inequality we divide both sides by $a$, but when $a<0$ we must reverse the inequality sign — the most common mistake. The result is given as an interval.

Special case $a=0$: the equation/inequality reduces to a condition on $b$ alone — it may always hold (an identity, $x\in\mathbb{R}$) or be contradictory (no solutions). Choose the relation from the list.