Absolute Value Equations and Inequalities

SAT Math · Axiom Academy

Absolute Value Equations & Inequalities Understanding distance and solving absolute value problems Absolute value measures distance from zero on a number line. Distance is always positive. Key insight: |x| can represent "distance from 0" or "distance from some point." When you solve |x| = 5 , you're asking: "What values are exactly 5 units from 0?" Rule: |x| = a creates TWO equations These are trickier. You need to think about "distance" carefully. Rule: |x| < a means distance from 0 is LESS than a This creates a range of values, not just two values. Rule: |x| > a means distance from 0 is MORE than a Absolute Value Inequality Summary Visual Representation on Number Line |x - 2| < 3 means distance from 2 is less than 3 Solution: -1 < x < 5 (between -1 and 5, not including endpoints) means distance from 2 is at least 3 Solution: OR (outside the region) Remember "distance": Absolute value is always about how far from a point. Less than = range, Greater than = two separate regions: This is the hardest part. Visualize it. Check negative values: When solving |x| = 5 , don't forget x = -5 is a solution. Symbol matters: use open circles (not included). and use closed circles (included).

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