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. Compound inequality: $c < ax + b < d$ (between two values) 7. Compound inequality with or: $ax + b < c$ or $ax + b > d$ 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, not just single values. 5. Example for case 1: Solve $2x + 3 > 0$. Step 1: Subtract 3 from both sides: $$2x + 3 > 0 \Rightarrow 2x > -3$$ Step 2: Divide both sides by 2 (positive, so inequality stays): $$x > \frac{-3}{2}$$ 6. The solution is all $x$ such that $x > -1.5$. 7. Each case follows similar steps but pay attention to the inequality direction when multiplying/dividing by negatives. This completes the explanation of the 7 cases of linear inequalities.