# 4: Inequalities

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## Learning Objectives

• Verbalize, visualize, and sketch number line graphs for inequalities.
• Translate compound inequalities to sketches and vice-versa.
• Find solutions to linear inequalities and visualize as number line graphs.
• Find solutions to absolute value inequalities and visualize as number line graphs.

• 4.1: Number Line Graphs
A real number line is a visual approach to ordering all real numbers. Any real number A plotted left of another real number B has the relation: A < B, or equivalently, B > A. We read aloud, “A is less than B” or equivalently, “B is greater than A.” The solution set to an inequality is the set of real numbers that make the inequality a true statement.
• 4.2: Interval Notation
Inequalities slice and dice the real number line into segments of interest or intervals. An interval is a continuous, uninterrupted subset of real numbers. How can we notate intervals with simplicity? The table below introduces Interval Notation.
• 4.3: Solving Linear Inequalities
Linear equations have just one solution. Linear inequalities have infinitely many solutions, requiring intervals to express solutions. To express solutions using set notation, the curly-brackets are used. The condition of the set is descriptive. The inequality describes the condition. A number must meet the condition to qualify as a solution.
• 4.4: Absolute Value Equations and Inequalities as Applied to Distance
The absolute value function, denoted y = |x|, takes any negative real number input and outputs the positive version of that number. Nonnegative numbers are left unchanged. Measuring distance is a good application to demonstrate the usefulness of this function. Distance is never negative.

This page titled 4: Inequalities is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jennifer Freidenreich.