How to Solve Inequalities: Step-by-Step Guide for Students

Understanding inequalities is a crucial skill in mathematics, especially as you prepare for assessments. Let’s dive deeper into how to solve inequalities step by step, using a dashed line graph as an example.

### Identifying the Type of Inequality
When graphing an inequality, the type of line you see is the first indicator of the inequality’s nature. A dashed line signifies that the inequality is either 'less than' (<) or 'greater than' (>), meaning the points on the line itself are not included in the solution. If the line were solid, it would indicate 'less than or equal to' (≤) or 'greater than or equal to' (≥).

In this example, the shaded region is below the dashed line, indicating that we are looking at a 'less than' inequality.

### Finding the Equation of the Line
Next, we need to express the line's equation in slope-intercept form, which is given as y = mx + b, where:
- **m** is the slope of the line.
- **b** is the y-intercept, the point where the line crosses the y-axis.

For our example, we can see from the graph that the y-intercept is at (0, 2). This means that when x is 0, y is equal to 2.

### Calculating the Slope
To find the slope, we need at least two points on the line. If we have the points (0, 2) and (2, 0), we can use the slope formula:

m = (y2 - y1) / (x2 - x1)

Inserting our points:
m = (0 - 2) / (2 - 0) = -2 / 2 = -1

This tells us that for every unit increase in x, y decreases by 1. Therefore, the equation of our line is:
y = -x + 2

### Writing the Inequality
Now that we have the equation of the line, we can express the inequality. Since the shaded area is below the line, the final inequality can be written as:
y < -x + 2

### Real-World Applications
Understanding how to solve and graph inequalities is not just about passing an exam; it's about developing critical thinking and problem-solving skills that are applicable in various real-world situations. For instance, inequalities can be used in budget planning, where you need to ensure expenses do not exceed a certain amount, or in physics, where they can represent limits on forces or distances.

### Conclusion
By breaking down the problem into manageable steps, you can approach inequalities with confidence. Practice with different equations and graphs to strengthen your understanding and prepare for your assessments. Remember, math is a skill that improves with practice, so keep training your brain!