Overview
This lecture explains the basics of inequalities, the meanings of different symbols, and how to represent inequalities graphically on a number line.
Inequality Symbols
- The < symbol means "less than" (e.g., x < 10 means x is any value less than 10).
- The > symbol means "greater than" (e.g., x > 10 means x is any value greater than 10).
- The symbols ≤ and ≥ mean "less than or equal to" and "greater than or equal to" respectively.
- Inequalities like x ≤ 10 include x = 10 as a possible value.
Reversing and Combining Inequalities
- x < 10 is the same as 10 > x; the values compared remain the same.
- Inequalities can include negative numbers (e.g., x < -5 means x is less than -5).
- Compound inequalities (e.g., 3 < x ≤ 7) mean x is between two values, possibly including one end.
Graphing Inequalities on a Number Line
- Use circles on a number line to show boundary values; fill the circle if the boundary is included (≤ or ≥).
- Leave the circle empty if the boundary is not included (< or >).
- Draw a line connecting the range of possible x values between the circles.
Worked Examples
- For 3 < x ≤ 7: circle at 3 (empty), circle at 7 (filled), connect between them.
- For -3 ≤ x < 4: filled circle at -3, empty circle at 4, connect between them.
Key Terms & Definitions
- Inequality — a mathematical statement comparing two values using <, >, ≤, or ≥.
- Less than (<) — the value on the left is smaller than the value on the right.
- Greater than (>) — the value on the left is larger than the value on the right.
- Less than or equal to (≤) — the value is either less than or exactly equal to another value.
- Greater than or equal to (≥) — the value is either greater than or exactly equal to another value.
Action Items / Next Steps
- Practice plotting different inequalities on a number line.
- Review and memorize the meaning of each inequality symbol.