1. The problem is to list and explain the 7 cases of linear inequalities. 2. Linear inequalities involve expressions like $ax + b > 0$, $ax + b \geq 0$, $ax + b < 0$, $ax + b \leq 0$, where $a$ and $b$ are constants and $x$ is the variable. 3. The 7 cases of linear inequalities are: 1. $ax + b > 0$ (strictly greater than) 2. $ax + b \geq 0$ (greater than or equal to) 3. $ax + b < 0$ (strictly less than) 4. $ax + b \leq 0$ (less than or equal to) 5. $ax + b \neq 0$ (not equal to zero) 6. $ax + b = 0$ (equal to zero, boundary case) 7. Compound inequalities such as $c < ax + b < d$ where $c$ and $d$ are constants. 4. Important rules: - When multiplying or dividing both sides of an inequality by a negative number, the inequality sign reverses. - Solutions to inequalities are ranges or intervals of values, not just single points. 5. Example: Solve $2x - 3 > 1$ Step 1: Add 3 to both sides: $2x - 3 + 3 > 1 + 3$ Step 2: Simplify: $2x > 4$ Step 3: Divide both sides by 2 (positive, so inequality sign stays): $\frac{2x}{\cancel{2}} > \frac{4}{\cancel{2}}$ Step 4: Result: $x > 2$ 6. This means all $x$ values greater than 2 satisfy the inequality. 7. Each case follows similar steps but pay attention to the inequality direction and whether equality is included.