1. The problem involves understanding and solving inequalities using a number line. 2. Inequalities are expressions that show the relationship between two values where one is greater or less than the other, such as $x > 3$ or $x \leq -1$. 3. To solve an inequality, we find the set of values for $x$ that make the inequality true. 4. For example, if the inequality is $x > 3$, the solution is all numbers greater than 3. 5. On a number line, this is represented by an open circle at 3 and shading to the right. 6. If the inequality is $x \leq -1$, the solution includes all numbers less than or equal to -1. 7. On the number line, this is shown by a closed circle at -1 and shading to the left. 8. When solving inequalities, remember that multiplying or dividing both sides by a negative number reverses the inequality sign. 9. For example, if $-2x > 6$, dividing both sides by -2 gives $x < \cancel{-2} \frac{6}{\cancel{-2}} = -3$. 10. Always check your solution by substituting values back into the original inequality. 11. The number line visually helps to understand the range of solutions. 12. This method applies to all linear inequalities and can be extended to compound inequalities.