Inequalities, Sequences, Functions, and Slope

Solving and Graphing Linear Inequalities

An inequality compares two mathematical expressions using relation symbols such as less than, greater than, less than or equal to, or greater than or equal to. When isolating a variable in an inequality, specific algebraic properties dictate whether the relation symbol stays the same or flips. Adding or subtracting the same number from both sides of an inequality preserves the direction of the inequality symbol. Similarly, multiplying or dividing both sides of an inequality by a positive number preserves the direction of the inequality symbol. In contrast, multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

Visualizing one-variable solutions on a one-dimensional number line involves distinct boundary markings and directional shading. On a one-dimensional number line, an open circle represents a strict inequality using the less than (<) or greater than (>) symbol, indicating that the boundary point is excluded from the solution set. A closed circle represents an inclusive inequality using the less than or equal to (≤) or greater than or equal to (≥) symbol, signifying that the boundary value satisfies the inequality. To represent the complete solution set, values less than a target boundary value are represented by shading to the left, while values greater than a target boundary value are represented by shading to the right. When two conditions are combined into a compound inequality, the connecting word determines the result: a compound inequality joined by 'and' represents the intersection of the solution sets of two individual inequalities, whereas a compound inequality joined by 'or' represents the union of the solution sets of two individual inequalities.

Graphing extends into two dimensions when working with inequalities in the Cartesian coordinate plane. In the Cartesian coordinate plane, the graph of a linear inequality containing < or > uses a dashed boundary line to show that points along the line are not solutions. Conversely, the graph of a linear inequality containing ≤ or ≥ uses a solid boundary line, showing that points on the line are included in the solution set. Because the boundary splits the plane into two regions, a test point not on the boundary line can be substituted into a two-variable inequality to determine which half-plane contains the solutions to be shaded. When multiple inequalities apply simultaneously, the solution set of a system of linear inequalities is represented graphically by the region where the shaded areas overlap.