Solve the inequality and graph the solution set. Write the...

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that involve a linear function and an inequality symbol (such as >, <, ?, or ?). They represent a range of possible values rather than a single solution. The process of solving a linear inequality typically involves isolating the variable on one side of the inequality, similar to solving linear equations, while keeping in mind that multiplying or dividing by a negative number will reverse the direction of the inequality.

Properties of Inequalities

The properties of inequalities are rules that govern the relationships between expressions when an inequality is established. These include adding or subtracting the same value from both sides without changing the inequality's direction, and multiplying or dividing both sides by a positive number without changing the inequality. Special attention is required when multiplying or dividing by a negative number because it reverses the inequality's direction.

Graphing Solution Sets

Graphing solution sets involves illustrating the range of values that satisfy an inequality on a number line. This type of graph typically uses open or closed circles to denote whether endpoints are excluded or included, respectively, and arrows to indicate that the set of solutions extends indefinitely in a particular direction.

Set-Builder Notation

Set-builder notation is a concise way of describing a set by stating the properties or conditions that its members must satisfy. In the context of solving inequalities, set-builder notation expresses the solution as the set of all variable values that make the inequality true.

Interval Notation

Interval notation is a method of writing subsets of the real number line by denoting the starting and ending points of the interval and whether these endpoints are included or excluded. It simplifies the representation of ranges of values by using parentheses for open intervals (endpoints not included) and brackets for closed intervals (endpoints included).

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