### Lesson: Inequalities and Interval Notation Recap

#### Introduction

Inequalities are like equations, but instead of equaling, they show that one side is greater or less than the other. They are essential for expressing ranges and conditions in various fields like mathematics, economics, and engineering. Interval notation is a way of writing sets of numbers, typically solutions to these inequalities. In this lesson, we’ll revisit how to solve inequalities and express solutions in interval notation, and we’ll explore online graphing tools to visualize these concepts.

#### Objectives

- Review the methods for solving linear and non-linear inequalities.
- Understand and practice writing solutions in interval notation.
- Use online graphing tools to visualize inequalities and their solutions.

#### Reviewing Inequalities and Interval Notation

**Solving Linear Inequalities**:- Similar to solving equations, but pay attention to the inequality sign (>, <, ≥, ≤).
- Remember to reverse the inequality sign when multiplying or dividing by a negative number.
- Example: Solve $2x+3>7$.

**Solving Non-Linear Inequalities**:- Involves inequalities with quadratic or higher-degree terms.
- Often require factoring and testing intervals to find the solution set.
- Example: Solve $x_{2}−5x+6>0$.

**Writing Solutions in Interval Notation**:- Express the solution as an interval that shows the range of values that satisfy the inequality.
- Use brackets [ ] for inclusive inequalities (≥, ≤) and parentheses ( ) for exclusive inequalities (> , <).
- Example: The solution $x>2$ is written as $(2,∞)$ in interval notation.

#### Utilizing Online Graphing Tools

Online graphing tools like Desmos or GeoGebra are excellent for visualizing inequalities:

##### How to Use

**Input the Inequality**: Enter the inequality into the graphing tool.**Visualization**: The tool will graph the inequality, shading the area where the inequality holds true.**Experiment with Different Inequalities**: Change the inequality to see how the graph and shaded area change.**Interval Notation**: Use the graph to understand the range of the solution set and express it in interval notation.

#### Exercise

**Inequality Challenges**: Solve a set of both linear and non-linear inequalities. Write the solutions in interval notation.**Graphical Interpretation**: Use a graphing tool to input these inequalities and observe the shaded solution areas. Confirm that your interval notation matches the graph.**Real-World Scenarios**: Create real-life problems that involve inequalities, like budget limits or measurement ranges. Formulate these into inequalities, solve them, and use the graphing tool for visualization.

#### Conclusion

Understanding how to solve inequalities and express solutions in interval notation is crucial for interpreting and solving a wide range of problems. By utilizing online graphing tools, you not only get a visual understanding of these concepts but also enhance your ability to accurately solve and represent inequalities. This skill is not only useful in academic mathematics but also in real-world scenarios where conditions and limitations are often expressed through inequalities. Keep practicing, and you’ll find these concepts increasingly intuitive and applicable in various contexts.