Absolute Value Without the ± Cases: Graph the V

Splitting |x − 5| into two cases invites sign errors in both. The graph shows the whole answer at once: a V you can look at.

The move: Graph the V, then just look.
Why it works: Absolute value is distance from zero, and its graph is a V — where the V sits (touching, crossing, or floating above the axis) IS the answer to solution-count questions.

The pattern

1. Type the absolute-value expression
2. Graph the V shape
3. Tap or count what it asks

Worked example: How many solutions does |x − 3| − 2 = 0 have?

1. Type the left side as a graph. That V shape is the absolute value. y=|x-3|-2
2. “= 0” means: where does it cross the x-axis? Tap the curve — two gray dots, at x = 1 and x = 5.
3. Two crossings → two solutions. You SAW the answer instead of reasoning about cases.

Worked example 2: What is the sum of the solutions of |2x − 1| = 7?

1. Graph both sides. y=|2x-1|, y=7
2. Two crossings: x = −3 and x = 4.
3. −3 + 4 = 1. The “±” case-splitting happened visually.

Remember it on test day

Graph the V, just look. That's the whole cue — say it when you see this question type and the steps follow.

When this isn't the move

Pure definition questions (“|−7| = ?”) are faster in your head than in any calculator.

Drill it until it's reflex

This move is one of Desvantage's 18 training modules: a plain-English lesson, a type-it-yourself walkthrough inside a live Desmos calculator, and a 100-question drill bank that deals ten fresh questions per round. The first two modules — plus the 12-question diagnostic — are free.