Inequalities, Factoring and Quadratics

Solving and Graphing Inequalities

Mathematics Knowledge is one of the four subtests, alongside Arithmetic Reasoning, Word Knowledge, and Paragraph Comprehension, that determine the AFQT score. Minimum qualifying scores are set by each military branch and can change over time. Because calculators are not permitted on the ASVAB, pencil-and-paper arithmetic and estimation are essential skills for solving algebra problems accurately under test conditions.

An inequality compares two mathematical expressions using specific relational symbols. The symbol < denotes 'less than', while the symbol > denotes 'greater than'. The symbol ≤ denotes 'less than or equal to', and the symbol ≥ denotes 'greater than or equal to'. Solving an inequality follows steps similar to solving a standard linear equation, but with an essential rule regarding negatives: adding the same real number to both sides of an inequality preserves the inequality symbol direction, and subtracting the same real number from both sides also preserves the direction. Multiplying or dividing both sides of an inequality by a positive number preserves the inequality symbol direction as well. However, multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality symbol.

Inequality solutions are displayed visually on a number line to show all allowable values. On a number line, a strict inequality (< or >) is represented at the boundary value by an open circle, showing that the exact cutoff point is not included in the solution set. An inclusive inequality (≤ or ≥) is represented at the boundary value by a closed solid circle, indicating that the boundary point satisfies the condition. Inequality solutions requiring values greater than a boundary point are graphed by shading to the right on a number line, whereas solutions requiring values less than a boundary point are graphed by shading to the left.

You will also see compound inequalities, which combine multiple conditions into a single statement. A compound inequality joined by 'and' represents the intersection of two conditions, requiring both to be true simultaneously. In contrast, a compound inequality joined by 'or' represents the union of two conditions, requiring at least one to be true. Graph an 'and' statement by keeping only values that satisfy both conditions (for example, x > 2 and x < 5 gives 2 < x < 5), and graph an 'or' statement by including values that satisfy either condition; depending on the conditions, the result need not be a bounded segment or two rays.