Inequalities C41758

1. Let's start by understanding what inequalities are. Inequalities compare two values and show if one is less than, greater than, less than or equal to, or greater than or equal to another. 2. The common inequality symbols are: $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), and $\geq$ (greater than or equal to). 3. When solving inequalities, similar rules to equations apply, but with an important exception: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. 4. For example, consider the inequality $2x + 3 > 7$. 5. To solve it, subtract 3 from both sides: $2x + 3 - 3 > 7 - 3$ which simplifies to $2x > 4$. 6. Next, divide both sides by 2 (a positive number), so the inequality sign stays the same: $\frac{2x}{2} > \frac{4}{2}$. 7. This simplifies to $x > 2$. 8. If instead we had $-2x + 3 > 7$, subtract 3 from both sides: $-2x > 4$. 9. Now divide both sides by $-2$, which is negative, so we reverse the inequality: $\frac{-2x}{\cancel{-2}} < \frac{4}{\cancel{-2}}$. 10. This simplifies to $x < -2$. 11. Always remember to reverse the inequality sign when multiplying or dividing by a negative number. 12. Inequalities can also be graphed on a number line or coordinate plane to visualize the solution set. 13. If you have a specific inequality problem, please provide it, and I can help solve it step-by-step.