Solve of these inequalities. Express the solution set in interval notation and graph it. See Exa

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Solve of these inequalities. Express the solution set in...

Key Concepts

Graphing on the Number Line

Graphing a solution on the number line is a visual representation of the set of numbers that satisfy the inequality. This involves plotting key endpoints using open or closed circles based on whether the endpoint is included or not, and then shading the region to the left or right of the endpoint to indicate all numbers that fall within the solution set.

Isolating the Variable

To solve a linear inequality, one of the primary steps is isolating the variable on one side of the inequality. This involves performing algebraic operations to move all terms containing the variable to one side and constant terms to the other. The goal is to express the inequality in a simplified form that clearly shows the range of possible values the variable can assume.

Interval Notation

Interval notation is a method of writing subsets of the real numbers that represent the solution of inequalities. It uses parentheses to denote that an endpoint is not included in the set (open interval) and brackets to denote that an endpoint is included (closed interval). This notation provides a concise and clear representation of all numbers that satisfy the inequality.

Linear Inequalities

A linear inequality is similar to a linear equation but uses inequality signs (<, ?, >, ?) instead of an equals sign. It represents all possible values of the variable that make the inequality true. Solving these involves similar steps as solving linear equations, such as combining like terms and isolating the variable, but with special attention paid to the direction of the inequality.

Properties of Inequalities

The properties of inequalities include operations like addition, subtraction, multiplication, and division. It is important to note that while addition or subtraction of the same number on both sides keeps the inequality consistent, multiplying or dividing both sides of an inequality by a negative number requires reversing the direction of the inequality sign.

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