SAT Absolute Value: Equations, Inequalities, and Graphing Absolute Value Functions

Absolute value |x| equals the distance from zero. When solving |x|=5, you have two cases: x=5 or x=-5. This two-case structure appears in every absolute value problem. Forgetting one case is the most common error. The systematic approach: identify the expression inside the absolute value, set up both cases (positive and negative), solve each, and check whether both solutions satisfy the original constraint.

Understanding absolute value as distance (not just the definition) helps you visualize problems and catch errors. Absolute value equations always have zero, one, or two solutions depending on whether the equation is possible within the distance constraint.

For |ax+b|=c, set up two cases: ax+b=c and ax+b=-c. Solve each case separately, then verify both solutions work in the original equation. For |2x-3|=7: case one (2x-3=7) gives x=5; case two (2x-3=-7) gives x=-2. Verify both: |2(5)-3|=|7|=7 ✓ and |2(-2)-3|=|-7|=7 ✓. Both work.

Practice 10 absolute value equations using the two-case method. Build the habit of checking both cases and verifying solutions. Common mistake: solving both cases correctly but forgetting to check that both actually satisfy the original equation.

The graph of y=|x| is a V shape with vertex at the origin. Transformations shift and stretch this V: y=|x-h|+k shifts the vertex to (h,k). y=a|x| stretches the V vertically by factor a. Recognizing these transformations helps you sketch graphs quickly without calculating every point.

Practice graphing five absolute value functions using transformations: y=|x-2|, y=|x|+3, y=2|x|, y=-|x|+1, and y=|x-1|-2. Build comfort with the V shape and its transformations. Spend one week on this skill until graphing becomes automatic.

For |x|<5, think "distance from zero is less than 5," which means -55, think "distance from zero is more than 5," which means x<-5 or x>5. This interpretation converts absolute value inequalities into compound inequalities automatically, avoiding the need to memorize rules. Practice converting and solving 10 absolute value inequalities using this approach.

Once you can solve absolute value problems systematically, build speed. Move from carefully working through each case to recognizing patterns and solving faster. Most students reach test-day speed within two weeks of daily practice on absolute value equations and inequalities.