Section 1.7 Linear Inequalities in One Variable

Review of Writing Sets Using Set-Builder and Interval Notation

In algebra, it is important to describe sets of numbers precisely. Two common notations used are set-builder notation and interval notation. These notations help express the solution sets of inequalities and equations.

• Set-Builder Notation: Describes a set by stating the property that its members satisfy. For example, the set of all real numbers less than 10 is written as .

• Interval Notation: Uses intervals to describe sets of numbers. The set of all real numbers less than 10 is written as .

• Graphical Representation: On a number line, an open circle at 10 indicates that 10 is not included in the set.

Example: The set can be represented in interval notation as and graphed with an open circle at 10, shading all values to the left.

Describing Solutions to Inequalities

Unlike equations, which often have a finite number of solutions, inequalities typically have infinitely many solutions. Solutions to inequalities can be described in three main ways:

• Graphing on a Number Line: Visual representation of the solution set.

• Set-Builder Notation: Expresses the solution set using a property.

• Interval Notation: Uses intervals to describe the range of solutions.

Objective 1: Solving Linear Inequalities

Definition and Properties

A linear inequality is an inequality that can be written in the form , where , , and are real numbers and . The inequality symbol can also be , , or .

• Solving Technique: Isolate the variable on one side using algebraic operations.

• Important Rule: When multiplying or dividing both sides of an inequality by a negative number, reverse the direction of the inequality symbol.

Example: Solve .

• Subtract 5 from both sides:

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

• Solution in set-builder notation:

• Solution in interval notation:

Objective 2: Solving Three-Part Inequalities

Definition and Solution Method

A three-part inequality (also called a compound or double inequality) involves a variable "sandwiched" between two numbers, such as . The solution is the set of values that satisfy both inequalities simultaneously.

• Solving Technique: Simplify the inequality so the variable is in the middle. Ensure the smaller number is on the left.

• Solution Criteria: A number is a solution if it satisfies both inequalities.

Example: Solve .

• Subtract 1 from all parts:

• Divide all parts by 2:

• Solution in set-builder notation:

• Solution in interval notation:

Summary Table: Notation for Solution Sets