Inequalities and Their Graphs

Key Concepts

1. What are Inequalities?

Inequalities are mathematical expressions that show the relationship between two values using symbols like <, >, ≤, and ≥. Unlike equations, which use the equal sign (=), inequalities show that one value is greater than, less than, greater than or equal to, or less than or equal to another value.

2. Types of Inequalities

There are four main types of inequalities:

• Greater Than (>): Represents that the left-hand side is strictly greater than the right-hand side.
• Less Than (<): Represents that the left-hand side is strictly less than the right-hand side.
• Greater Than or Equal To (≥): Represents that the left-hand side is greater than or equal to the right-hand side.
• Less Than or Equal To (≤): Represents that the left-hand side is less than or equal to the right-hand side.

3. Graphing Inequalities on a Number Line

Graphing inequalities on a number line helps visualize the range of values that satisfy the inequality. The steps to graph an inequality are:

1. Identify the critical point (the value where the inequality might change from true to false).
2. Use an open circle (○) for strict inequalities (< and >) and a closed circle (●) for inclusive inequalities (≤ and ≥).
3. Shade the number line to the left or right of the critical point, depending on whether the inequality is less than or greater than.

Examples and Analogies

Example 1: Graphing x > 3

To graph the inequality x > 3:

1. The critical point is 3.
2. Use an open circle at 3 because it is not included in the solution set.
3. Shade to the right of 3 to indicate all values greater than 3.

Example 2: Graphing x ≤ -2

To graph the inequality x ≤ -2:

1. The critical point is -2.
2. Use a closed circle at -2 because it is included in the solution set.
3. Shade to the left of -2 to indicate all values less than or equal to -2.

Analogy: Temperature Range

Think of an inequality as a temperature range. For example, if the temperature must be at least 10°C, you can represent this as T ≥ 10. On a number line, you would place a closed circle at 10 and shade to the right, indicating all temperatures from 10°C upwards.

Conclusion

Understanding inequalities and their graphs is crucial for solving real-world problems and visualizing mathematical relationships. By mastering these concepts, you can better interpret and solve problems involving ranges of values.