How to Solve Linear Inequalities: A Step-by-Step Guide

Solving linear inequalities is an essential skill in algebra that helps you understand relationships between variables. A linear inequality is similar to a linear equation, but instead of an equals sign, it uses inequality symbols (≤, <, ≥, >). Here’s a step-by-step guide on how to tackle these problems effectively.

### Step 1: Define Your Variables
Before beginning your calculations, it’s crucial to define your variables clearly. For instance, if you are trying to determine the number of yoga and spin classes, you could define your variables as follows: let y represent the number of yoga classes and s represent the number of spin classes. This will help you keep track of your calculations.

### Step 2: Write the Inequality
Next, you need to set up the inequality based on the context of the problem. In our example, the inequality is 5y + 10s ≤ 200. This means that the total number of participants in yoga and spin classes should not exceed 200. Each yoga class accommodates 5 people, and each spin class accommodates 10 people.

### Step 3: Substitute Values
Once you have the inequality, substitute in the values for y and s. If you want to check the feasibility of holding 6 yoga classes and 8 spin classes, substitute y = 6 and s = 8 into the inequality:
- Calculate the total number of participants from yoga classes: 6 yoga classes × 5 participants = 30 participants.
- Calculate the total number of participants from spin classes: 8 spin classes × 10 participants = 80 participants.
- Add the two results together: 30 + 80 = 110 participants.

### Step 4: Analyze the Result
Now, compare the total (110 participants) with the maximum allowed (200 participants). Since 110 is less than 200, the values you've chosen for y and s do satisfy the inequality. This step is crucial as it confirms that your choices are feasible within the constraints of the problem.

### Real-World Application
Understanding linear inequalities is not just an academic exercise; it has real-world applications in various fields. For example, businesses often use linear inequalities to determine production limits, budget constraints, and resource allocation. By mastering this concept, you can apply it to situations involving costs, profits, or even scheduling in everyday life.

### Conclusion
In summary, solving linear inequalities involves defining your variables, writing the inequality, substituting values, and analyzing the results. Always remember to clearly define your variables at the beginning of your solution. By following these steps, you can confidently tackle linear inequalities and apply them in practical scenarios. Keep practicing, and you will gain proficiency in this essential math skill!