| $$ \boxed{ax + b \gt 0} $$ |
| $$ \underline{\underline{\bullet \ a \gt 0}} $$ |
| $$ \begin{aligned} x &\gt -\frac{b}{a} \\ \\ x &\in \left(-\frac{b}{a},\ +\infty \right) \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a \lt 0}} $$ |
| $$ \begin{aligned} x &\lt -\frac{b}{a} \\ \\ x &\in \left(-\infty,\ -\frac{b}{a} \right) \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a = 0}} $$ |
| $$ \underline{\circ \ b \gt 0} $$ |
| $$ x \in \left(-\infty,\ +\infty \right);\ x \in \mathbb{R} $$ |
| $$ \underline{\circ \ b \le 0} $$ |
| $$ x = \left\{\right\} $$ |
| $$ \boxed{ax + b \ge 0} $$ |
| $$ \underline{\underline{\bullet \ a \gt 0}} $$ |
| $$ \begin{aligned} x &\ge -\frac{b}{a} \\ \\ x &\in \left\langle-\frac{b}{a},\ +\infty \right) \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a \lt 0}} $$ |
| $$ \begin{aligned} x &\le -\frac{b}{a} \\ \\ x &\in \left(-\infty,\ -\frac{b}{a} \right\rangle \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a = 0}} $$ |
| $$ \underline{\circ \ b \ge 0} $$ |
| $$ x \in \left(-\infty,\ +\infty \right);\ x \in \mathbb{R} $$ |
| $$ \underline{\circ \ b \lt 0} $$ |
| $$ x = \left\{\right\} $$ |
| $$ \boxed{ax + b \lt 0} $$ |
| $$ \underline{\underline{\bullet \ a \gt 0}} $$ |
| $$ \begin{aligned} x &\lt -\frac{b}{a} \\ \\ x &\in \left(-\infty,\ -\frac{b}{a} \right) \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a \lt 0}} $$ |
| $$ \begin{aligned} x &\gt -\frac{b}{a} \\ \\ x &\in \left(-\frac{b}{a},\ +\infty \right) \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a = 0}} $$ |
| $$ \underline{\circ \ b \lt 0} $$ |
| $$ x \in \left(-\infty,\ +\infty \right);\ x \in \mathbb{R} $$ |
| $$ \underline{\circ \ b \ge 0} $$ |
| $$ x = \left\{\right\} $$ |
| $$ \boxed{ax + b \le 0} $$ |
| $$ \underline{\underline{\bullet \ a \gt 0}} $$ |
| $$ \begin{aligned} x &\le -\frac{b}{a} \\ \\ x &\in \left(-\infty,\ -\frac{b}{a} \right\rangle \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a \lt 0}} $$ |
| $$ \begin{aligned} x &\ge -\frac{b}{a} \\ \\ x &\in \left\langle -\frac{b}{a},\ +\infty \right) \end{aligned} $$ |
| $$ \underline{\underline{\bullet \ a = 0}} $$ |
| $$ \underline{\circ \ b \le 0} $$ |
| $$ x \in \left(-\infty,\ +\infty \right);\ x \in \mathbb{R} $$ |
| $$ \underline{\circ \ b \gt 0} $$ |
| $$ x = \left\{\right\} $$ |
