Interval Notation

Understanding Interval Notation

Interval notation is a concise way to describe sets of real numbers, especially those defined by inequalities. It is commonly used in precalculus to represent solution sets for inequalities.

• Open Interval (a, b): All real numbers between a and b, not including the endpoints.

• Closed Interval [a, b]: All real numbers between a and b, including the endpoints.

• Half-Open Intervals (a, b] or [a, b): Include only one endpoint.

• Infinite Intervals: Use ∞ or −∞ to indicate unbounded intervals. Infinity symbols are always paired with parentheses, not brackets.

Examples: Writing in Interval Notation and Graphing

• a) Interval notation: [1, 3]

• b) Interval notation: (−4, 0)

• c) Interval notation: (5, ∞)

• d) Interval notation: (−∞, 1]

Properties of Inequalities

Basic Properties

Solving inequalities relies on several fundamental properties, which allow us to manipulate and solve for the variable:

1. Addition Property: If , then for any real number .

2. Multiplication Property (Positive): If and , then .

3. Multiplication Property (Negative): If and , then . Note: Multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

Additional info: These properties also apply to the other inequality symbols (, , ) in the same way.

Linear Inequalities

Definition and General Form

A linear inequality is an inequality that can be written in the form , where . The inequality symbol can be , , , or .

• To solve a linear inequality, use the properties of inequalities to isolate the variable.

• The solution is often expressed in interval notation and can be represented graphically on a number line.

Solving Linear Inequalities: Examples

• a)

• b)

• c) (Compound inequality)

• d) (Compound inequality with fractions)

To solve these, follow these steps:

1. Isolate the variable on one side of the inequality.

2. If you multiply or divide by a negative number, reverse the inequality sign.

3. Express the solution in interval notation.

4. Graph the solution on a number line if required.

Example Solution:

• Subtract 3 from both sides:

• Divide both sides by -2 (reverse the inequality):

• Interval notation:

Example Solution:

• Add 2 to all parts:

• Divide all parts by 3:

• Interval notation:

Additional info: Compound inequalities are solved by performing the same operation on all three parts of the inequality.